Ordinary Differential Equations: A Brief Eclectic Tour (Classroom Resource Materials)

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Ordinary Differential Equations A Brief Eclectic Tour

David A. Sanchez

Texas A&M University

Published and distributed by

The Mathematical Association of America

CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary classroom material for students-laboratory exercises, projects, historical information, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. Committee on Publications

Gerald Alexanderson, Chair Zaven A· Karian, Editor David E. Kullman Frank Farris Millianne Lehmann Julian Fleron Sheldon P. Gordon William A. Marion Stephen B Maurer Yvette C. Hester William J. Higgins Edward P. Merkes Judith A. Palagallo Mic Jackson Andrew Sterrett, Jr. Paul Knopp 101 Careers in Mathematics, edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus Mysteries and Thrillers, R. Grant Woods Combinatorics: A Problem Oriented Approach, Daniel A. Marcus Conjecture and Proof, Mikl6s Laczkovich A Course in Mathematical Modeling, Douglas Mooney and Randall Swift Cryptological Mathematics, Robe� ifM i:l'fEUl d Elementary Mathematical Models�lt.f O, F(x 1 ) < 0, least one solution in the interval x0 < x < x1.

xo

1 then it is < t < oo, whereas if I CI :5 1 it will only be defined on a finite interval. =

defined for -oo For instance

x(1r /2) = � implies C = 2, x(t) (2 - cos t)-3, x(1rj2) = 8 implies C 1/2, x(t) = � - cos t -3 , =

=

which is defined only for

2

1r /3

g' (x0)

< 0.

Then given any solution

xo but sufficiently close to Xo,

hence

x(t)

ddx.t /

t=to

=

g (x1 )

is decreasing near

we have

0, K = carrying capacity > 0 ,

K (stable). The model is discussed in with (unstable), and x 2 many of the current textbooks, as well as the variation which includes harvesting, first discussed by Fred Brauer and the author in a 1 975 paper in the journal Theoretical Population Biology. That equation is

dx dt

=

rx

=

( 1 - Kx ) - H,

H = harvesting rate

> 0,

and a little calculus and some analysis shows that as H increases from 0, the unstable equilibrium increases from 0, and the stable equilibrium decreases from K. When H reaches r KI 4 they both coalesce and for H > r KI 4 we have !� < 0 for all x so the population expires in finite time. For devotees of bifurcation theory the behavior can also be analyzed from the standpoint of a cessation of stability when the parameter H reaches the critical value r KI 4: a saddle-node bifurcation. One can now pile on more equilibria and look at equations like

dx dt

=

(x - x1 ) (x - x 2 )

·

·

·

(x - Xn )

and put them on the computer and look at direction fields, a pleasant, somewhat mindless exercise. Things can get wacky with equations like

19

2 First Order Equations

dx

- =

1

·

sm -

,

"

WI

· · fimtte · number of eqm"l"b 1 na X n th an m

X & converging to 0 and alternating in stability, or

=

1

± - , n = 1 , 2, . . . , �

d(}

= s i n2 (} , with an infinite number of semistable equilibria X n = 0, ± 1r, dt d(} . . . ± 21r, . . . , smce 1 num m . . the vtctmty . of any eqm"l"b > 0. dt The last example will come up later in the discussion of the phase plane. The case where the � ght-hand side of the differential equation depends on t and x (called the nonautonomous case) is much trickier, and while there are some general results, the analysis is usually on a case by case basis. The following example is illustrative of the complexities.

Example.

dx dt

=

t (1

- x2 )

x(1 + t2 )

, x =/: 0.

We see that x1 (t) = 1, x 2 (t) = - 1 are solutions, so by uniqueness all other solutions lie :>etween - 1 and 1 , are below - 1 or are above 1. Now we must do some analysis:

Case: 0 < l x l

< 1.

Then

:

dx - > 0 if dt dx ->0 dt

�ase: l x l >

1.

Then

��

if

0 when t = 0 and

{x>O {x

if

dx dt

if

dx

t>0

- < 0 if dt dx

t0

=

=

- < 0 if dt 0 and

{ x>1 { -1 t>O

dx - < 0 if dt

t>0 X

We see that the two solutions agree at t = 1 , but

'

dx'

dx 7 e2 ' = 7e 2 + 2. dt = l + dt = l t t The second expression for x( t ) can also be obtained by solving x x(l) = �e 2 + �· =

= 2x

+ 1,

Finally, i n keeping with thi s book' s philosophy that the flfSt order equation i s an excellent platform to introduce more general topics at an elementary level, the author would like to discuss briefly the subject of singular perturbations. This will be done via a simple example which gives a glimpse into this fascinating topic of great interest in applied mathematics. For a more in-depth discussion see the references by Robert O'Malley. Consider the IVP f±

where f.

>

=

-x + 1 + t, x(O) = 0

0 is a small parameter.

The solution is

x(t )

=

( f. - l)e - t / E + t + (I - f. )

and we see that it has a discontinuous limit as I.lffi

E -+0+

X

(t) -

{ 1+ 0,

f. �

0+

t t, t

=

>

0 0

Furthermore, if one is familiar with the "big 0" notation, then

( )

. = �1 - 1 e-t / € + 1 x(t)

�

1

� for 0

the needed estimate we will use one of the workhorse tools of nonlinear ODE'ers. Its proof can be found in almost any advanced textbook and is not difficult.

u(t) and v(t) are nonnegative continuous functions on u(t) :S a + 1tot v(s)u(s) ds, t to,

Gronwall' s Lemma: If

to t :S

< oo,

and satisfy

2:

where a is a nonnegative constant, then

u(t) :S a exp [1: v(s) ds] , t to . We will assume f ( t, x) satisfies the conditions for E and which implies that there exists a constant K 0 such that the Lipschitz condition, i f (t,x) - f(t,y) l :S K l x - y i , is satisfied in some neighborhood of (t0, x o ). The constant K could be an upper bound on I� I, for instance. Then each solution satisfies xi (t) xo + 1tot f(s,xi (s)) ds, i = 1 , 2 , where Xo = a for x 1 (t), and xo b for x 2 (t), and we are led to the estimate l x 1 (t) - x2 (t) l I a - b l + 1tot lt(s,x l (s)) - f (s, x2 (s)) I d!J. 2:

U,

>

=

=

:S

39

2 First Order Eq uations

Applying the Lipschitz condition gives

! x 1 (t) - x2 (t) I 'S I a - bl + 1tot K ! x 1 (s) - x2 (s) l ds and then Gronwall' s Lemma with Ia - bl, u(t) ! x 1 (t) - x 2 (t)l, and v(t) K, results in the needed estimate a =

=

=

which requires s_ome commentary. First of all, the estimate is a very rough one, but does lead us to the conclusion that if Ia - b l is small the solutions will be close initially. There is no guarantee they will be intimately linked for t much greater than nor is the estimate necessarily the best one for a particular case. Example. If

for t
2.3025. The important fact we can conclude is that solutions are continuous functions of their initial values recognizing this is a local, not global, result.

w-6

10

t

w-4

t

Continu ing Continuatio n

The proof of the E and U Theorem depends on the method of successive approximations, and some fixed point theorem, e.g., the contraction mapping theorem. Given the IVP, x let = and compute (if possible)

= f(t, x), x(to ) = xo , xo (t) xo . x 1 (t) = xo + lt f(s,xo(s)) ds, x2 (t) = xo + lt f(s, xl (s) ) ds, . . . Xn+ I (t) = Xo + 1tot f(s,xn (s) ) ds, �

�

,

etc. This is a theoretical construct; after a few iterations, the integrations usually are colossal if not impossible. You want approximations? Use a numerical scheme. A general set of conditions under which the above successive approximations will converge to a unique solution of the IVP are as follows. First, find a big domain (open, connected set) B in the -plane in which and are continuous-the proof is the same for a system where E Next, construct a closed, bounded box

tx

f(t,x) 8f(t,x)j8x x Rn .

40 r

Ordinary Differential Equations-A Brief Eclectic Tour

{ ( t, x) I it - to I

=

S: a,

lx - xo I

$ b} contained in B-you can make this as big as

you want as long as it stays in B. Since

r is closed and bounded, by the properties of continuity we know we can find positive numbers m and k such that if(t, x) l S: m, l8f(t, x) j8xl $ k for (t, x) in r. It

would be nice if our successive approximations converged to a solution defined in all of

r, but except for the case where

f(t, x) is linear this is usually not the case. r > 0 satisfying the three constraints

To assure

convergence we need to find a number

r

r

S: a,

Then, the successive approximations

x( t)

of the integral equation

x(t) for

it - t0 1

r.

S:

=

x(t)

Furthermore,

$ b/m ,

Xn (t), n xo +

r

and =

0, 1 ,

k

< 1/ .

2, . . .

converge to a unique solution

i. f (s, x(s)) dx , t

to

is a unique solution of the

IVP.

But what could happen is that the t-dimension of our rosy-eyed choice of the box may have been considerably reduced to attain an interval on which a solution exists.

r - - - - - : - (� :.� - �7: - - : �x(t) �

B

l.

r

-

- -

r

--:

:

�------�---------�-------: 2r 2a

However, looking at the picture above, it appears that we can construct a new box B with center value

r1

( to

+

r, x ( t0

+

r)),

and start a new

IVP there. We would create a new

and by glueing the two solutions together get a solution of our original

o - r $ t $ to (t o - r, x(to - r)) as well.

defined for t

+

r + r1 .

r 1 in

IVP now

We could do some carpentry at the other end

The continuation theorem says we can continue this process as long as we stay in B,

insure that

f is bounded in B, and the right- and left-hand limits of the solution exist and

are in B as we hit the right- and left-hand boundaries, respectively, of the box r. The

process continues and the solution is defined for all t, or on some infinite half interval, or it becomes infinite and fails to exist for some finite value of

t. ( t0 , xo),

The discussion implies that associated with each initial value spondjng solution of the

an d the corre

IVP is a maximum interval of existence which is the largest ,

interval on which the solution is defined. It can be thought of as the ultimate finale of the continuation process described above. It is generally impossible to compute except when the

IVP can be explicitly solved.

However, the successive approximation scheme applied to a linear equation has one very nice consequence:

41

2 First Order Equations

��

Given

=

a(t)x + b(t),

a(t)

where

r1

then every solution is defined for

and

a,

h = b]:/ , N

select a step size

of the solution

x(t)

of the

NP ±

=

To

f(t, x), x(a) xo, =

a positive integer, then the two methods are expressed by

the simple "do loops":

Euler's Method:

t0 = a xo = xo

for n from

0

to

N

-

1

do

tn+ l = t n + h Xn+t Xn + hj(tn , Xn ) tN b!) XN =

(it better be

print

print

Improved Euler's Method:

to = a xo = xo

for n from

0

to

N

-

1

do

Un Xn + hj(tn, Xn) Xn+ l Xn + 2h [J (tn , Xn ) + J (tn+ l • Un)] tN XN =

=

prin t print

They are easily understood routines, but it is surprising how some expositors can clutter them up. Modifications can be added such as making a table of all the values n =

0,

.

.

.

, N,

the step size

3

h.

Xn x(tn), �

or plotting them, or comparing the values or plots for different values of

Key Stuff

What are the important points or ideas to be presented at this time? Here are some:

The Geometry:

All numerical schemes essentially are derivative chasing or derivative

chasing/correcting/chasing schemes, remembering that we only know one true value of

±

(t),

that for

to a =

where

± (to ) f(t0, xo). =

For Euler' s Method we use that true

45

3 Insight not Numbers

x 1 x(tl), x(t 2 ) . etc.

value to compute an approximate solution value of

± (tl )

�

f(t1 , xi), then the next value x2

�

X

slope

x2 ----------

0

slope

h

h t

l

then an approximate value

�

f(t

f(t 0 ,

1 , x1

x0

)

)

'

t

2

For the Improved Euler Method we compute the approximate value of using Euler' s Method, then use the average of approximation of

f(t 1 , xl ),

x1

�

x(tl ),

f(t0, xo)

f(tl o x 1 )

± (h ) � f(tl o x1 )

to get a corrected

then compute the new approximate value of

use that to get an approximate value of

correct, etc. The picture is:

x(t2 )

slope

X

new x 1

X

and

�

j(t 2 , x 2 ).

----

0

to

�

l

2 (m o + m l ) ) m1

slope f(t 1, x l

l

± (t 1 )

average again and

slope f( t 0 , "o

=

)

=

II\)

A further nice insight is to note that for the case where the ODE is simply x f(t), x(a) x0 then x(t1 ) = x0 + A1 where A1 i s th e area under th e curve f(t), to � t � t1 • Euler ' s Method uses the left-hand endpoint rectangular approximation to that area, =

=

whereas the Improved Euler Method uses the trapezoidal approximation, which is more accurate. Cost :

The Euler Method is a one step method requiring only one function evaluation for

each step, whereas the Improved Euler Method requires two. The classical Runge-Kutta Method requires four. For problems where the solution is very wiggly or gets large, and consequently a very small step size

h

and lots of function evaluations are needed to try

to obtain accurate approximations-this comes with a cost. If that cost, which is really a measure of how much number crunching must be done, is too great the little gnome inside the computer might j ust trash the computation.

46


Error: The important error of a numerical method is the global error which determines the order of the method, but almost equally as important is the local error. Suppose we are given the

NP ± = f(t, x), x(t0)

= x0,

have selected a step size

h,

and have

generated a sequence of points with the method :

Xo = x(to), x l � x(ti ) , . . . , xn � x(tn), tk+l = t k + h, k = O, l, . . , n - 1 . .

l xn - x(tn ) J , and the method is said to be of order r O ( hr ) , which means there is a constant, dependent on things like

The global error is the value if

Jxn - x(tn) l

=

f,

the initial value and the function Euler Method is of order

4.

Runge-Kutta is of order

l xn - x(tn ) ! :S M hr .

such that

For instance the

1, the Improved Euler Method is of order 2, and the classic

The local error is defmed in terms of one step of the iteration as follows: numerical scheme has landed you at point value problem

(tj+I , Xj+I )·

i = f ( t, x), x( tj ) = xj

(tj , xi)

where

Xj � x(tj )·

suppose the

For the

new initial

compute one step of the method to get the point

The local error is the value

f g is the global error and fe the local one:

Jx(tJ+I ) - Xj+I i ·

A picture is helpful where

X

- - - - - - - - -

- - - - - - - - - - -

In general, if the global error is of order

r

: }Eg

·

the local error is of order

r

The important points to b e emphasized are i)

If the numerical method is of order

with new step size

h

=

hjm,

r,

and we compute with step size

where say

m

we expect the global error to be reduced by size then

l xn - x(tn) l

:S Mhr

implies

+ 1.

2, 4 , 1 0 , 10 2 , 1/mr . If XN �

=

J xN - x(tn) l :S M

•

•

•

, 10 23

x(tn)

h,

then again

(your choice),

for the new step

(�) = ; ( r

hr .

The last remark has several implications. It is the basis of a reasonable test of the accuracy of the calculations. Compute with step size could equal

2,

or

4,

or

h,

then again with step size

h/m

where

m

10, for instance. Decide beforehand the accuracy you need, in

terms of significant figures, and compare the two answers. Use their difference to decide

47

3 Insight not Numbers

whether you need to recompute with a smaller step size or can stop and go have a latt�. The second implication i s i i ) D o not expect that if you keep reducing the step size you will get closer and closer to the solution. The fiendish roundoff error may contaminate your results; that is usually spotted by heartwarming convergence as the step size is successively reduced, then at the next step the answers go wacky. Also do not expect that even with a very small step size and a reasonable order method you will always get a close approximation. The constant

M

in the error estimate could be huge, or the method, which computes

its own directions, may have gotten off the mark and is merrily computing a solution far removed from the actual one.

4

RKF M ethods

This section is not intended as a tutorial on RKF methods (the RK stand for Runge-Kutta, the F for H. Fehlberg who first developed the theory for the method), but to familiarize the reader with the underpinnings of the method. It' s not a bad idea to know what the little gnome inside the computer is up to. The foundation of the method is based on the strategy that by carefully controlling the size of the local error, the global error will also be controlled.

Furthermore, a general

procedure for estimating the size of the local error is to use two different numerical procedures, with different orders; and compare their difference. Fehlberg was able to do this using Runge-Kutta methods of order 4 and 5, so the local errors

O(h5 )

and

O(h6 ) ;

are

respectively

his method involved six function evaluations at each step.

This was incorporated into a computing scheme which allows for changing the step size

h

by following this strategy:

Select�ng a step size

h,

estimate the relative (to

h)

size of the local error by using the two method procedure described above; this gives a number l est. l . Next, decide what is an allowable relative local error f, and now use the following criteria in computing i ) If l est. l > f reject the size.

ii) If l est. I < f, accept

Xk+ l

Xk+ l

(tk+ l , Xk+ d

given

(tk , Xk):

obtained, and compute a new

and compute

xk+2

Xk+ l

using step size

Of course, in clas sroom practice the parameters l est. ! ,

t: ,

h

h

using a smaller step

or a larger one.

and others are already built

into the RKF45 program so you will see only the computations. Some calculators use an RKF23 program. The advantages of the RKF45 scheme besides its inherent accuracy are that when the solution is smooth, and the approximations are closely tracking it, the step size selected can be relatively large, which reduces the number of function evaluations (cost ! ) . But if the solution is wiggly the step size can be reduced to better follow it-a fixed step size method could completely overshoot it. Furthermore, if the solution is too wild or flies off in space the RKF methods have a red flag which goes up if l est. l < f cannot be attained.

For schemes which allow

for adjustments, this requires a bigger value of f be given, otherwise for schemes with preassigned parameters the little gnome will fold its tent and silently steal into the night.

48

Ordinary Difterentlal Equations-A Brief Eclectic Tour

Given the discussion above does the author believe that students should become facile or even moderately familiar with Runge-Kutta methods of order 3 or 4?

Absolutely Not! They require hellishly long tedious calculations or programming because of the number of function evaluations needed to go one step, and why bother when they are available in a computer or calculator package.

Use the Euler and Improved Euler

Methods to introduce the theory of numerical approximation of solutions of the

IVP, and

the notion of error. Then give a brief explanation of RKF45 , if that is what is available,

so they will have an idea of what is going on behind the computer screen. Then develop insight.

5

A Few Examples

The ideal configuration is to have a calculator or computer available to be able to do the Euler and Improved Euler Methods, then have a numerical package like RKF45 to check for accuracy and compute approximations far beyond the capacity of the two methods. Start off with a few simple IVPs to gain familiarity with the methods .

Comment: The author finds it hard to support treatments which present a large number of problems like: For the initial value problem

x = 2x , x(O)

a) Use the Euler Method with

h

=

x(1).

�, k

b ) Use the Improved Euler Method with value of

=

1

1 and 1 6 to approximate the value of

h

=

x(1).

c) Find the solution and compute

x(1),

�, k. and

1 1 6 to approximate the

then construct a table comparing the

answers in a) and b). This is mathematical scut work and will raise the obvious question find the exact solution why in blazes are we doing this?"

"If we can

Here are six sample problems-given these as a springboard and the admonitory tone of this chapter the reader should be able to develop many more (insightful) ones. We have used RKF45 .

Problem 1 (straightforward). Given the

IVP

x = v'x + t, Use the Euler Method with approximate

x(1)

h

=

0.05

x(O)

=

3.

and the Improved Euler Method with

h

=

0. 1

to

then compare the differences with the answer obtained using RKF45 . Euler,

h

Improved Euler,

h

Answers:

RKF45 :

=

0.05 : x ( 1 )

=

0. 1 : x ( 1 )

x(1)

�

�

�

5.0882078

5 . 1 083323

5 . 1 089027

As expected, Improved Euler gave a much better approximation with twice the step size. The equation can be solved by letting

x=

u2

-

t, but one obtains an implicit solution.

49

3 Insight not Numbers The next three problems use the old veteran faster than tan t.

x

x2 + t2 ,

=

whose solutions grow much

Problem 2. A strategy to estimate accuracy, when only a low order numerical method is in your tool kit, is to compute the solution of an again with step size

h/2,

x and approximate

x(1)

NP

using a step size

h,

then compute

then compare results for significant figures. Do this for =

x 2 + t2 ,

using a step size

h

=

x(O) 0.2

0

=

and

h

=

0. 1

with the Improved Euler

Method. Answer:

h = 0.2 : x( 1) ;:::::; 0.356257 h

At best, we can say

=

0 . 1 : x ( 1 ) ;:::::; 0.351830

x ( 1 ) ;:::::; 0.35.

Problem 3. Given the threadbare computing capacity suggested in Problem 2 you can improve your answer using an interpolation scheme. Given a second order scheme we know that if size

h,

x(T)

is the exact answer and

xh (T)

is the approximate answer using step

then

a) Show this implies that

1 4 x(T) ;:::::; 3 xhf (T) - 3 xh(T) 2 2 to obtain RKF45.

b) Use the result above and the approximations you obtained in Problem improved estimate of

Answer:

a)

x(1),

then compare it with that obtained using

x(T) - xh (T) ;:::::; Mh2 x(T) - x h/ (T) ;:::::; M ( � ) 2 2

�

=

�

i Mh2

b) x ( 1 ) ;:::::; (0.35 1830) - (0.356257) RKF45 Problem

4. For the

NP x

=

gives

x2 + t2 , x(O) = 0.

Euler,

h

=

=

(

x T)

0.350354

Approximate the value of

h

(graphically if you wish) with the answer obtained with

h

solve for

x ( 1 ) ;:::::; 0.350232

the Euler and Improved Euler Methods and step sizes

Answer: Euler,

=

}

x(2)

0 . 1 and h = 0.01. RKF45. =

0. 1 : x(2) ;:::::; 5.8520996 0.01 : x(2) ;:::::; 23.392534

(Now you suspect something screwy is going on) Improved Euler,

h = 0. 1 = x(2) ;:::::; 23.420486

(Maybe you start feeling confident) Improved Euler,

h

an

=

0.01 : x(2) ;:::::; 143.913417

RKF45: x(2) ;:::::; 317. 724004

(Whoops !)

with

Compare

50

Ordinary DIHerential Equations-A Brief Eclectic Tour

The awesome power of RKF45 is manifest! The next problem uses the direction field package of MAPLE, possessed of the ability to weave solutions of initial value problems through the field, to study the existence of periodic solutions of a logistic equation with periodic harvesting. Problem 5. Given the logistic equation with periodic harvesting

x

�x_ ( - �) - ( � 1

=

a) Explain why it has a (stable) 2-periodic (T

=

+

� sin 1rt)

2) solution.

x(O) x(2). Use MAPLE direction field plots and x(O) then graph the solution. Compute x(4)

b) Such a solution must satisfy = RKF45 to find an approximate value of as a check. Discussion:

l � t + l sin 1rt < i t + l sin 1rt ) > 0. For ( !

we see that for x = 2 and all t, x = x 4 and all t, x = 0 - (t + l sin 7rt) 0, so by an argument given in Ch. 2 there exists a 2-periodic solution x(t) with 2 x(O) 4. b) Construct a direction field plot for 0 � t � 2, 2 � x 4, and select a few initial conditions x(O) = 2. 4 , 2.8, 3.2, 3 .6, and plot those solutions. You get something like -- -- -- -- ----" ---" ---" ---" ---" ---" ---"---" ---" ---"---" ---" a) Since

.2 . Then x1 (t) e>-1 t and x 2 (t) e>- 2 t are solutions and t t W(t) det (>.::� 1 t >.: ::2 t ) = (>.2 - >.1)e ( >.1+>. 2 ) t '::/: 0 , =

=

=

=

=

=

so they are a fundamental pair.

p(>.)

e>-1t

b. has double root >.1 , so x1 (t) = is one solution-what is the other? One can guess x 2 (t) = and find out that it works-definitely a time saving approach! But this is a nice time to reintroduce reduction of order, an important tool. First note that if is a double root of then from the quadratic formula >.1 - � and b = a4 .

>.1

te>-1 t

p(>.)

=

2

65

4 Second Order Equations

x2 (t) = eA1t t t u(s)ds and substitute it into the differential equation to get, after eA1 , + (2.X 1 + a)u = 0. AteA1 t + BeA1 t . The second term is This implies = 0 so u(t) = A, and x 2 (t) original x 1 (t), and since A is arbitrary we conclude that x 2 (t) = teA1 t is the second solution. Then W( t) = det ("'I'eA1eAt1 t eA1 t +teA"'I'1 tteA1 t ) = e2A1 t =f. 0, and therefore x 1 (t) and x2 (t) are a fundamental pair. c. p(.X) has complex roots, and since b are real they must be complex conjugate pairs .X1 = r + ifJ , .X 2 = r - ifJ, where i 2 - 1 . Now let canceling the

u

u

our

=

a,

=

Even if the neophyte has not studied complex variables, it i s essential that formula be introduced at this point:

an

important

ei8 = cos (} + i sin (}. It can be taken for granted, or motivated by using the infinite series for cos(} and sin fJ, and the multiplication rules for products of i. Besides, it leads to the wonderful relation ei1r + 1 0, if (} = 1r, wherein e, i, 1r, 0 and 1 share in a fascinating concatenation. Letting (} ---. -fJ gives e -ifJ = cos(} - isinfJ and you obtain the key formulas cosfJ = eifJ +2 e-ifJ , sin(} = eifJ -2ie - ifJ The holds so x1 (t) = eA1 t = e ( r +i fJ ) t = er t e i8t , and x 2 (t) = e A 2t 8 (e r-ilaw fJ) t oferexponents t e- i t , are solutions, and any linear combination x(t) c1x1 (t) + c2 x2 (t) is also a solution. Nothing about linearity prohibits c 1 and c2 being complex numbers so frrst let c 1 c2 = �: XI (t) = -21 ert etOt. + -21 ert e -•fJ. t = ert ( eifJt +2 e -ifJt ) = ert cos (Jt, a real solution ! Now let c1 �. c2 = - � and obtain X2 (t) = -2i1 e2t et8t. - -2i1 ert e -•fJ. t = ert ( eifJt _2ie -ifJt ) = ert sinfJt , =

=

=

=

=

=

another real solution ! Now compute their Wronskian

since

W(t) = (rert cosfJertt cos- fJer(Jt t sinfJt rert sin(Jertt +sinfJer(Jtt cosfJz) = fJe2rt (cos2 (Jt + sin2 fJt) = fJert =f. 0

(} =f. 0; they are a fundamental pair.

66


In

these few pages we have covered everything the beginner needs to know about

linear, homogeneous (no forcing term), constant coefficient equations, which should be reinforced with some practice. Further practice should come from doing some applied problems.

Remark. The Euler Equation

.. a . b x + tx + 2 x t

=

t

0'

# 0,

should be briefly mentioned, since it is a camouflaged linear equation and does come up in some partial differential equations, via separation of variables. It also provides absurdly complicated variation of parameters problems for malevolent instructors. Solution tech nique:

let

x (t)

=

t\

= ..\ 2 + (a - 1) ..\+ b,

substitute to get a characteristic polynomial p(..\)

then work out the three possible cases as above.

Remember, the Euler Equation is one of the few explicitly solvable fly specks in the universe of nonconstant coefficient linear differential equations. Do you want to venture beyond

n=

2? You do so at your own risk in view of the

fact the majority of high school algebra texts have shelved factoring polynomials in the same musty bin they stored finding square roots. Of course your friendly computer algebra

have to wave your

program can do it for you but what is being learned? And then you will

hands and assert the resulting solutions form a fundamental set of solutions because they are linearly independent, unless you want to demonstrate some sleep-inducing analysis.

Example 1. Wish to prove e>-1 t , e>. 2 t , . . . , e>-n t, Ai distinct, are linearly independent? You could write down their Wronskian

W(O) =

det

(;

W(t),

then

,

n 1 ..\ 1 -

Ai

which is Vandermonde ' s determinant (Whoa ! ) and is never zero if the

are distinct. Or

suppose -oo
.1t,

< oo,

e-(>.t ->.)t, differentiate . . . , eventually get something like Acne(>.n ->-n- t )t = 0 where A depends on all the differences and is not zero so en 0. Now repeat to show Cn-1 = 0, etc . , and the

ci

t

differentiate, then multiply by

=

making sure to drop a book o n the floor once i n awhile.

Example 2. It gets worse when you have multiple roots ..\1 so e>-; t, te>-; t , . . . ,

tke>.; t

are solutions if

..\1

has multiplicity

k.

A

linear combination of all the solutions

being identically zero will produce an expression like -oo
.1 t , and differentiate enough p1 (t) disappear. Now multiply by e- < >. 2 ->. t )t , differentiate some more, . . .

where the

times to

make

. Maybe

you

are a little more convinced of the wisdom of sticking with

n

=

2.

67

4 Second Order Equations 7

What To Do With Solutions

To start with an elementary, but very useful, fact, it is important to be able to construct and use the phase-amplitude representation of a periodic sum: If ¢(t) a sin wt + b cos wt then ¢(t) = A cos(wt - ¢) where A = va2 + b2 is the amplitude and ¢ tan- 1 alb is the phase angle or phase shift. =

=

This is trivially proved by multiplying the first expression by Ja2 + b2 I Ja2 + b2 and using some trigonometry. The representation makes for a quick and illuminating graphing technique:

A = J34 and ¢ = tan- 1 315 ¢(t) = v'34 cos (2t - 0.54), and 2t - 0.54 = 0 gives t 0.27.

Example. ¢(t) = 3 sin 2t + 5 cos 2t so

�

0.54 rad., hence

=

J34

- 54

We will shortly see the effective use of the representation in studying stability. There are three popular models from mechanics used to discuss undamped or damped motion; we first consider the former: / /

(i)

/

/ /

/

,

- - - - 1'-.- -

: L I

v_

_ _

_

(iii)

(ii)

x=O

�n

L X

x small m

(i) is a suspended mass on a spring, (ii) is the mass connected to a spring moving on a track, and (iii) is a bob or pendulum where x is a small angular displacement from the rest position. In (i) L is the length the spring is stretched when the mass is attached. All share the same equation of motion

x + w 2 x = 0,

the harmonic oscillator,

where w2 gIL in (i) and (iii), and w 2 = k, the spring constant in (ii). In the case of the pendulum the equation of motion is an approximation for small displacement to the true equation, x + w 2 sin x = 0. Most problems are easy to solve except for the penchant some authors have for mixing units-the length is in meters, the mass is in drams, and it takes place on Jupiter. Stick with the CGS or MKS system ! =

68

Ord inary Differential Equations-A Brief Eclectic Tour

If there is a displacement mass, then the NP is

x0

from rest

x + w 2x

whose solution is

=

0,

:

x

0, or an initial velocity

=

x(O)

=

Xo, x(O)

=

Yo

V

Yo

imparted to the

-

x(t) sin wt + Xo cos wt = x5 + Y5 /w 2 cos(wt ¢) ¢ = tan- 1 y0jwxo. The phase-amplitude expression gives us a hint of the phase =

where plane as follows:

x(t) and therefore

Or if we plot

x(t)

vs.

=

}

- w x5 + y5fw 2 sin(wt - ¢)

x(t) = y(t) then x(t) 2 y(t) 2 + x5 + y5fw 2 w5x5 + Y5

1

'

which is an ellipse. It is the parametric representation of all solutions satisfying x(to) Furthermore, we see from the phase-amplitude representation of the solution that a change in the initial time is reflected in a change in the phase angle of the solution, not the amplitude. If we add damping to the model, which we can assume is due to air resistance or friction (in the case of the pendulum at the pivot), and furthermore make the simple assumption that the dissipation of energy is proportional to the velocity and is independent of the displacement, we obtain the equation of damped motion

x0, x(to ) = y0 for some t0•

=

c > 0. Now things get more interesting; the standard picture demonstrating the model might add a dashpot to represent the frictional force:

Models of RLC circuits can also be developed which lead to the same equation, but since the author doesn't know an ohm from an oleaster they were omitted. The motion depends on the nature of the roots of the characteristic polynomial-you could imagine it as the case where the cylinders of your Harley Davidson' s shock absorbers were filled with Perrier, Mazola, or Elmer's Glue. The polynomial is p(>.) = >. 2 +c>. +w 2 with roots

>. 1 , 2 =

� ( -c ± v'c2 - 4w2 ) i ( -1 ± J1 - �2 ) . =

69


4w 2

Case 1 . Overdamped, 0

< cr < 1 .

Both roots are distinct and negative: At -rt . A 2 = -r2 , and x(t) r1 , r2 > 0 , where a and are detennined b y the IC. If rt < r2 then =

b

which goes to zero as

= ae - r1t + be-r2 t,

t ---t oo. The solutions will look like this: X

Remark. Most plots depicting this case look like the one above, but the solution can have a small bump before plummeting to zero. Look at x(t) 3e-t - 2e- 2 t. =

Case 2. Critically damped, � = 1 . There is one double root A = -c/2 < 0 and the solution satisfying is If x0, y0

to zero.

x( t ) = xoe- ct / 2 + ( Yo + cxo/2)te- ct f 2 .

are

positive it will have a maximum at t

= t c = 1c

(

Yo

x(O) = xo, ± (0) = Yo

/0cx9 72

)

>

0 then decrease

X

Since the size of the maximum could be of interest in the design of the damping mechanism, assigned problems should be to find the maximum value of x(t) and when it occurs. Case 3. Underdamped,

4w2

cr >

1.

Now the roots are - � ± iO , ()

=

J� - 1 , so the solution will be

x(t) = ae - ctj 2 sin ()t + be - ct f 2 cos ()t = Ae - ct/ 2 cos(Ot - ¢)

where the maximum amplitude A and the phase angle conditions. A possible graph would be:

are

detennined by the initial

70


X

The dashed

graph is

± A(cos cll ) e-ct/2

0

The interesting problems for the underdamped case are to estimate a ftrst time beyond which the amplitude is less than a preassigned small value for all later times. These problems afford a nice combination of analysis and the advantages of today' s calculator or computer graphing capabilities. Examples.

1 . For the system governed by the

1

NP

.. . x + 8 x + x = 0 , x(O) = 2, x(O) = o,

l x (t) l

accurately estimate the smallest value of T for which

< 0.5 for all

t :2: T.

Discussion: This can be done solely by using a graphing calculator or a plotting package, but doing some analysis ftrst has benefits. The solution is

x(t)

�

2.0039e-t / l6 cos

and since cos 0 has local extrema at (} = mr, n 16(mr+O. o626) . Then we want mr to get t n = J255

(¥255

)

16 t - 0.0626 ,

=

0, 1 , 2, .

l x(tn ) l = 2. 0039 exp( -tn )

.

.

, ftrst solve

< 0.5

which gives a value of n � 7. Check some values: n = 7, n = 8, n = 9,

tn tn tg

22.097, 25.2 45, � 28.392, �

�

x(t7) x(ts) x (tg)

� � �

-0.5036 0. 4 136 -0.3398

So we get this picture: 0.5

0

--0.5

X

v[p" t - 0.0626 =

71


( v;;

Now call in the number cruncher to solve

2.0039e- t 11 6 cos test

w i th

A

2.

t7

=

::::::

t-

0.0629

)

=

-0.5

22.097 to get T ::::::: 22. 16968423 an d x (T) ::::::: -0.49999999.

similar problem is the following: Find a second order linear differential equation and initial conditions whose solution is

x ( t)

=

2e- t 1 20 sin t.

for which

Then find an accurate estimate of the smallest value of

i x (t) i :

Then

=

and the bracketed term will have a large amplitude and a large period term will have a period

"'

so we see the phenomenon of

211'

/E.

The other

beats, where the large

amplitude, large period oscillation is an envelope which encloses the smaller amplitude,

smaller period oscillation.

Case 2:

q=1

Instead of using undetermined coefficients, or variation of parameters for the intrepid souls, to solve this case, go back to the first expression for the solution in the case

q =f 1,

78


and let

q= 1

+ f.. We have

x(t) = 1 - ( F1 ) 2 [ s( 1 )t - s t] + f.

co

+ f.

co

and with a little trigonometry and some readjustment this becomes

sinf.t ] . . xE (t) -F [(cost) COSf.t - 1 - (smt)-=

2

-

Now use L' H(')pital ' s Rule letting ·f. ---> lim

E-+0

which is the dreaded case of

f.

f.

+ f.

0 to get

xE(t) x(t) = F t sin t, 2

=

resonance.

X

t

0

For the more general case x +

problem might

A

be

w2x = Fcoswt

we get

x(t) = i:,tsinwt.

A

sample

one similar to those suggested for the underdamped system:

system governed by the

IVP

x x = -16 cos t, x(O) x(O) = 0, l x (t)l ..

+

4

2

"blows up" when the amplitude

A

=

7r

exceeds 24. When does this first occur?

more general discussion of resonance can

be

obtained for the constant coefficient

case by slightly adjusting the fonn of the variation of parameters fonnula, or using the following result: The solution

x(t)

of the IVP

x + ax +

bx = q(t), x(O) = x(O) = 0

is given by the convolution integral

xp(t) = 1t ¢(t - s)q(s) ds ¢(t) ¢ ¢(0) = 0, (0)

where

is the solution of the homogeneous differential equation satisfying =

1.

79


xp(t): xp(t) = <jJ(t - t)q(t) + 1t ¢(t - s)q(s ) ds = 1t ¢(t - s )q( s) ds since ¢(0) 0, xp(t) = ¢ (t - t)q(t) + 1t ;fi(t - s)q(s) ds q(t) + 1t ;fi(t - s)q(s) ds since ¢ (0) = 1.

To prove the result, first compute the derivatives of

=

=

Then

xp + axP + bxp = q(t) + 1t (;fi + a¢ + b<jJ) (t - s)q(s) ds q(t) since ¢( t) is a solution of the homogeneous equation. =

A more elaborate proof would be to use the variation of parameters formula for each of the constant coefficient cases and combine terms. The convolution integral representation has several very nice features:

xp(O) = xp(O) = 0, x + ax + bx = q(t), x(O) = Xo, x(O) = Yo easier to solve. One merely adds to xp(t) the solution of the homogeneous problem

a. Since the particular solution contributes nothing to the initial conditions, which makes the IVP

satisfying the IC.

b. Similar to the case for the first order equation it gives an elegant representation of the solution when is a discontinuous function.

q( t)

Example.

x. - 2x. + x = { -11 : 20 � tt

=

=

f(t), will have a 21r-periodic solution for all t. Otherwise, there will be resonance.

The result allows us to consider a variety of functions

f(t)

satisfying

f(t + 21T)

for example a)

10 211" (sin(t - s))l sin s i ds = 10.,. sin(t - s) sin s ds 211" 1 + sin(t - s) (- sin s) ds = 0,

f(t) = l sin t i ,

then

.,.

s o n o resonance

=

f(t) ,

81


b)

f(t) = { -� 7l"0 (O) cf>(t - s) = cf>(t)cf>- 1 (s) t s. sin(t - s)f(s) ds = 0 sint 127r coss f(s)ds - cost 127r sins f(s)ds 0, for all t. particular, this would be true for t = /2 and t = which implies that resonance will =

In

7r

7r

not occur if the orthogonality relations

{ 2 7r sinsf(s) ds = O s s f(s) ds 0, = Jo Jo { 2 7r co

are satisfied.

10

Second Thoughts

There are some other facets of second order, linear, differential equations which could be part of the introductory course, and they merit some comment.

Numerical Methods

For second order equations numerical methods have no real place in a first course, assuming that the usual (hopefully not too long) introduction to the topic was given when first order equations were discussed. What should be made clear is that despite what the computer screen might show, the numerical schemes convert the second order equation to a first order system--remember they can only chase derivatives. For example, to use Euler's Method for the general IVP

x = f(t,x,x), x(a) = xo, x(a) = yo,

it is first converted to a first order system

x = y, iJ = f(t,x,y), x(a) = xo , y(a) = Yo· Then the algorithm to approximate the values x(b), y(b) = ± (b) is:

82


Euler's Method

to = Xo = xo Yo = Yo a

for n from 0 to N

-

1 do

tn+l tn + h Xn+l = Xn + hyn Yn+l = Yn + hj(tn , Xn , Yn ) =

where

h = b!t .

Once this is explained, leave the numerics to the little gnome and enjoy his sagacity.

Unless the presentation is intended as preparation for a course in partial differential equations, and consequently the emphasis is on linear differential equations, there will be little time for boundary value problems. The Sturm Liouville problem, self-adjointness, eigenfunction expansions/Fourier series etc. are topics far too rich to cover lightly. But this does not mean the subject should not be mentioned, if for no other reason to point out the big differences between the initial value problem and the boundary value problem. The big distinction is of course that we are requiring the solution to satisfy end point conditions at two distinct points as opposed to one initial point. A simple example points this out: Boundary Value Problems

Given the equation y" + y = 0, where y = y(x), how many solutions does it have for the following boundary conditions? (i) y(O) = 1 , y ( ! ) = 1: since y(x) = A sin x+B cos x is the general solution, then we must solve y(O) y

=

A sin(O) + B cos(O) = 1

(�) = A sin (�) + B cos(11'/2)

=

so y (x) = sin x + cos x is the unique solution.

1

}

� A = B = 1•

(ii) y(O) = 1 , y(11') = 1 : we obtain y(O) = A sin(O) + B cos(O)

=

B

=

1,

y(11') = A sin(11') + B cos(11') = - B = 1 , a contradiction, so no solution exists.

83


(iii)

y(O) = y(27r) = 1: we obtain A sin(O) + B cos(O) 1 } B 1, A arbitrary A sin(27r) + B cos(27r) 1 so we have an infinite number of solutions y(x) = Asinx + cosx. for boundary value problems, the independent variable is usually x (space) as =

=

=:::}

=

Note: opposed to t (time). Another problem worth mentioning is the eigenvalue problem; an example is this: Given the boundary value problem

y" + >.y = 0, y(O) = y(1r) = 0, for what values of the parameter >. will it have nontrivial solutions? The example indicates that the question of existence and uniqueness of solutions of a boundary value problem is a thorny one.

-r2 , r 0, the general solution is y(x) = Ae-rx + Berx . Then y(O) = 0 implies A + B = 0, and y(1r) = 0 gives y(1r) = Ae- r1r + ( -A)er1r = 0. A = 0 we have the trivial solution y(x) :::::: 0, otherwise A(e -r1r - er1r) = 0 gives 1 e2r1r 0, which is impossible. No solutions. ii) >. = 0: then y" = 0 or y(x) = Ax + B and the boundary conditions imply that A = B = O. y(O) = 0 iii) >. 0: let >. r2 , r 0, then y(x) = Asinrx + Bcosrx; B = 0, and y(1r) sinr1r 0 implies >. = 1,4, . . . ,n2 , , the of the differential equation with associated sin x, sin 2x, . . . , sinnx, . . . . i) >.

If

0:

Letting >.

>

=

=

=:::}

>

=

=

=

• • •

ues

eigenval

eigenfunctions

This is a good point to stop, otherwise one is willy nilly drawn into the fascinating topic of Fourier series. Remark. If you are driven to do some numerical analysis, taking advantage of today's accessible computer power, consider doing some numerical approximations of boundary value problems using the shooting method. It has a nice geometric rationale, and one can use bisection or the secant method at the other end to get approximations of the slope. The book by Burden and Faires has a very good discussion. Some sample problems:

i)

y" + e2xy' + y 0, y(O) 0, y(1) = 0.5 =

=

(initial guess ii)

y" + 2y' + xy = sinx, y(O) = 0, y(1) 2

y' (0) 1.2) =

=

(initial guess

y' (0) = 4.0)

84

iii)


y" - £ y5 = 0 , y(O) = 2 / 3 , y( 7I 4)

=

1 /2

(initial guess

y' (O) = -0. 1 5)

Stability This is a good spot to briefly mention the stability of the solution x(t) = 0 of the constant coefficient equation x + ax + bx 0. Going back to the mass spring system we see that if x(t) = x(t) = 0, the system is at rest-this is an equilibrium point. From the nature of solutions we see that =

a. If the roots A of the characteristic polynomial have negative real parts then every solution approaches 0 as t - oo. The cases are

AI , A 2 < 0 : x(t)

=

CI €>. 1t + c2 e>.2 t

ci e>.t + c2 te>. t A = r ± io: , r < 0 : x(t) = Aert cos(o:t - cp).

A < 0 double root: x(t)

=

each case w e see that for any € > 0 i f lx(t) - Ol i s sufficiently small for t lx(t) - 01 < € for all t > t1 o and furthermore lim lx(t) l = 0. We conclude that

In

=

ti then

t -+oo

If the roots of the characteristic polynomial have negative real parts then the solution x(t) = 0 is asymptotically stable. b. If the roots of the roots A of the characteristic polynomial are purely imaginary A = ±io:, then the solution is x( t) = A cos( at - cp) . We see that if A, which is always positive, satisfies A < €, then lx(t) - O l = I A cos(o:t - cp) l ::; IAI < €, so as before, if lx(t) - Ol is sufficiently small for t = h then lx(t) - Ol < € for all t > ti, but lim lx(t) l :;6 0.

t -+oo

We conclude that If the roots of the characteristic polynomial have zero real parts then the solution

x(t)

=

0 is stable.

From the argument above we see that for a solution to be asymptotically stable it must first be stable-an important point that is sometimes overlooked. c. Finally in the case where one or both of the roots of the characteristic polynomial have a positive real part we say that the solution x( t) = 0 is unstable. We can use a default definition that not stable means unstable, and clearly in the cases AI , A 2 > 0, double root A > 0, or A = a + i(3, a > 0, we see that any solution x(t) becomes unbounded as t - oo no matter how small its amplitude. What about the case AI < 0 < A2 ? While there is a family of solutions CI e>-1 t which approach zero as t - oo, the remainder are of the form c2 e>-2t or ci e>-1t + c2 e >. 2t which become unbounded. Given any € > 0 we can always find solutions satisfying lx(O) I < € where x(O) = c2 or x(O) = CI + c2 and c2 # 0. Such solutions become unbounded as t - oo so the homespun stability criterion�nce close always close-is not satisfied.

85


It is important to go through the cases above because they are an important precursor to the discussion of stability of equilibria in the phase plane of a linear or almost linear two-dimensional autonomous systems.

In fact,

they serve as excellent and easily analyzed

models of the various phase plane portraits (node, spiral, saddle, center) of constant coefficient systems. This will be demonstrated in the next chapter.

Infinite Series Solutions Perhaps the author' s opinion about power series solutions is best summarized by a footnote found in · Carrier and Pearson' s succinct little book on ordinary differential equations, at the start of a short referred to is an

ten page chapter on power series. The Chapter 1 1 eleven page chapter with some of the important identities and asymptotic

and integral representations of the more common special functions-Error, Bessel, Airy, and Legendre.

sin x, sin x I sin xi 1, sin x sin(x + 27r)

* For example, the trigonometric function,

is an elementary function from

the reader' s point of view because he recalls that smooth function for which

:::;

is that odd, oscillatory,

=

and for which

meticulous numerical information can be found in easily accessible tables; the Bessel function,

J0(x),

on the other hand, probably won ' t be an elementary

function to that same reader until he has thoroughly digested Chapter

J0(x) x 1,

11.

Then,

will be that familiar, even, oscillatory, smooth function which, when

»

is closely approximated by

differential equation,

�

(xu')' + xu 0, =

cos(x /4), J0(0) = 1, -

for which

1r

which obeys the and for which

meticulous numerical information can be found in easily accessible tables. The statement was made in ber

1 968 when practicing mathematicians used tables (remem tables?), long before the creation of the computing power we enjoy today. So why

do we need to have books with

30-50

page chapters of detailed, boring constructions

of power series for all the various cases-ordinary point, regular singular point, regular singular point-logarithmic c ase-followed by another lengthy presentation on the power series representations of the various special functions ? 1

Unless you intend to become

a specialist in approximation theory or special functions, you are not likely to have to compute a series solution for a nonautonomous second order differential equation in your entire life. Somebody long ago, maybe Whittaker and Watson, decided that the boot camp of ordinary differential equations was series solutions and special functions, including the Hydra of them all, the hypergeometric function.

This makes no sense today and the

entire subject should be boiled down to its essence, which is to be familiar with how to construct a series solution, and a

recognition of those special functions which come up

in partial differential equations and applied mathematics via eigenfunction expansions. The point made in the quote above is an important one, namely that there are differential equations whose

1

explicit solutions are special functions, whose various properties such

The author labored long and hard, seeking enlightenment, writing such a chapter in a textbook some time

ago. Whatever nascent interest he had in the topic was quickly throttled.

86


as orthogonality and oscillatory behavior are well known, and whose representations are convergent power series with known coefficients. For instance:

The Bessel function of order 2 J2 (x) =

( x ) 2 00 ( - 1 ) n ( x ) n 2 � n!(n + 2) ! 2

is a solution of the Bessel Equation of order 2, d2 y 1 dy x2 - 2 2 + -- + 2 y 2

dX

X dX

X

=

0.

The function J2 ( x) is an oscillatory function with an infinite number of zeros on the positive x-axis, and J2 (x) ----> 0 as x ----> oo. It is well tabulated . Hence, we can certainly say that we can solve the Bessel equation of order 2, although the solution is not in terms of polynomials or elementary functions. Power series solutions do give us a broader perspective on what is the notion of the solution of an ODE. Introduce the topic with the construction of the first few terms of the series for Ai(x) or J0 (x), then develop the recurrence relation for their terms. For the same reason you should not use Euler' s Method to approximate the solutions of x x, do not develop the series expansion for the solutions of x + x = 0; it gives the subject a bad reputation. Next, if you wish, discuss briefly some of the special functions, their differential equations, and possibly some of the orthogonality relations, but this is already wandering far afield. Then move on, remembering that if someone wants the graph of J7; 2 (x) or its value at x = 1 1 .054 a computer package or even some hand calculators will provide it-but who would ask such a question? Something worth considering instead of power series expansions is perturbation expansions; these are of great importance in applied mathematics and in the analysis of periodic solutions of nonlinear ODEs. See for instance, the Poincare-Lindstedt method in more advanced texts. First order equations can provide some nice examples as the following shows: =

Consider the logistic equation with periodic harvesting

x

=

1 x ( 1 - x/40) 10

(� + E sin t) .

Then r = 110 , K = 40 so r K/4 = 1, the critical harvesting level, so for existence of periodic solutions we require that I � + E sin t l < 1 or l E I < !· We wish to approximate the stable periodic solution with a power series in E having periodic coefficients, so let

Substitute and equate like powers of E : 0 �. The only periodic solutions will be �: : xo = 110 xo - 460 x6 constant ones xo ( t ) = 10, 30. These correspond to the new equilibria under constant rate harvesting H = � . To perturb around the stable

-

87


xo (t) 30. f 1 ± 1 = ( 110 2�0 xo ) x 1 sin t = - 210 x1 - sin t. The solution is x1 (t) = A 1 e - t l 20 + ° (20 cos t - sin t) 01 0 cos(t + . is an eigenvalue of is an eigenvector corresponding to

x

>.e >.t "l

=

�

�

But where to go from here to get

>.. A one

=

A then �

e>.t "l

=

is a solution, where "1

line proof suffices :

Ae >. t "l e>.t (A - I>.) ry

=

0.

�

.i with associated eigenvectors i· i = 1, 2 , . . . , n then the set { � i (t) } = {e>.; t ryi } is a � fundamental set of solutions of the system � A� . =

(iii) We can quickly take care of the case n

=

2 completely, and it gives a hint of what

the problem will be for higher dimensions when multiplicities occur.

(iv) The subj ect is ordinary differential equations not linear algebra. Recently, some textbooks have appeared doing a lot of linear algebra for the case n

=

3

and then discussing three dimensional systems.

from the availability of better

3-D

The impetus for this comes

graphics, and because of the great interest in the

three dimensional systems exhibiting chaotic behavior-the Lorenz equations modeling climate change, for instance. Unfortunately, two-dimensional systems cannot have chaotic structures, but that isn ' t a good reason to spend excessive time analyzing Returning to the cozy n = 2 world:

consider

3 x 3 matrices.

and we can state:

>. 1 , >.2 are distinct � 1 (t) = e>.1 t 'fl l > � 2 (t)

If

eigenvalues with associated eigenvectors

ry 1 , ry2,

�

e>.2 t ry2

=

then

�

are a fundamental pair of solutions and every

solution can be written as

for appropriate choices of the constants

c 1 , c2.

What about the case of complex conj ugate eigenvalues >.

A, if >. is A,

For any real matrix "1 then since

A

=

=

r

+

iq, 5.

= r

- iq, q =f. 0?

a complex eigenvalue with associated complex eigenvector

(A - >.I)ry

=

0 (A

-

XI)i]

=

0,

91

5 Linear and Nonlinear Systems

so ij is an eigenvector associated with .X, and the statement above applies. But we want real solutions, so go back to a variation of the technique used for the second order equation whose characteristic polynomial had complex roots. If >. = r + iq and TJ = ( a + i(3, 1 + io) then any solution can be expressed as

[ ( i)

�

x (t ) _

�

ert c 1 ei qt

( )]

a - i(3 a + (3 iq i: + c2 e - t i: 1 - zu 1 + Zu .

.

•

Now let c 1 = c2 = � . then c1 = t;. c2 = - t;, and express cos qt, sin qt in terms of complex exponentials to get two real solutions x 1 (t)

�

x 2 (t )

=

-

�

( (

ert ert

a cos qt - (3 sin qt . I COS qt - O Sm qt a sin qt + (3 cos qt i: . 1 sm qt + u sm qt •

) ),

and there' s our fundamental pair. The case of >. a double root is vexing when we can only find one eigenvector linear algebra approach is to find a second nonzero vector Then a fundamental pair of solutions is

Xt (t)

=

e;>..t TJ,

x 2 (t )

=

,...._,

i"J

a

TJ.

(A - >.I)a

The

�

satisfying

�

�

=

TJ.

e;>..t (a + ,..._,ryt ) , ""

and this must be proved by showing � is not a multiple of � - We will do this shortly. But a differential equations approach is more intuitive, suggested by the strategy used t in the second order case of multiplying by t. Try a solution of the form t e;>.. ry it t"'oJ

-

doesn't work. So try instead that solution plus x (t ) =

f"V

te;>..t TJ + e ;>..t a f"V

gives

A ry = >.ry

is assumed to be a solution, and since

±

=

>.te;>..t TJ + e;>..t ( TJ + >.a) ,..., I'V

,..._,

=

k(A - >.I) �

£=�

a

=

a

is a nonzero vector. Then

± = Ax , a little calculation

,..._,

I"V

te;>..t A TJ + e;>..t A a. f'V

I"V

so the first terms cancel and after cancelling

which is exactly what we want. If a were a multiple of TJ, say I'V

where

,...._,

""'

But

� e ;>..t

�

e;>..t

kry, then the equations

we are left with

(A -

>.I )kry

=

TJ

would

which is impossible. Now we can prove � 1 (t ) and � 2 (t) are a fundamental pair of solutions by showing we can use them to solve uniquely any IVP, ! A � , �(0) col (xo , Yo) with a linear combination imply

,...._,

=

=

=

x (t ) = f"V

Ct Xt (t) + C2 X 2 (t) ""

f"V

If TJ = col(u, ) a = col(w, ) v ,

�

,...._,

�

z

=

Ct e).. t ,..._,TJ + C2 e).. t (a + try ) .

this leads to the equations

f"V

I'V

""

""

92

Ordinary DiHerentlal Equations-A Brief Eclectic Tour

which can be solved for any

x0 , y0 since the columns of the coefficient matrix ( � � )

are

linearly independent, so its determinant is not zero.

All this takes longer to write than it does to teach, but limiting the discussion to n

=2

makes the calculations easy and avoids all the problems associated with dimensionality. Just consider n

=

3 with ..\ an eigenvalue of multiplicity three and the various possibilities ;

you might shudder and just leave it to the linear algebra computer package. For the nonhomogeneous system

�

=

A(t) � + � (t)

�

or

=

A� + � (t)

we have discussed the general variation of parameters formula for n

= 2 in the previous A note that the chapter. It generalizes easily to higher dimensions. In the case A(t) variation of parameters formula does not require the computation of - 1 ( s ) when ( s) is the fundamental matrix satisfying (O) = I. This is because (t) - 1 (s) = (t - s) =

which is not hard to prove:

A then O(t) n(t) = ( t ) - 1 ( s ) and since � � (t) - 1 ( s ) = A(t) - 1 (s) = An(t). Therefore n(t) is the solution of the IVP X = AX, X ( s ) = I, but so is (t - s ) hence by uniqueness n(t) = ( t - s ) . Let

=

=

Finally we note that the method of comparison of coefficients can be used to solve

� = A � + � (t) if � (t) has the appropriate form, but it can be a bookkeeping nightmare.

For example to find a particular solution for the system

(iJx) = A (xy) (sint2 t) +

{ Yxp(t) Bt2 + Ct + + E sin t + F cos t p(t) = Gt 2 + Ht + J + K sin t + L cos t D

=

try

substitute and compare coefficients, and that is assuming of the solutions of

�

=

A � . Good

sin t is not one of the components

luck-if you do the problem three times and get the

same answer twice-it' s correct.

2

The Phase Plane

The phase plane is the most valuable tool available to study autonomous two-dimensional systems

x = P(x, y) iJ

=

Q(x, y),

x and y-there is P = P(x, y, t), Q = Q(x, y, t) the

and i t i s important t o stress that the right-hand sides depend only o n no t-dependence. If there is t-dependence so that

system can be transformed to a 3-dimensional autonomous system by introducing a third

variable

z = t.

Then we have

which sometimes can

x = P(x, y, z) , iJ = Q(x, y, z ) , i = l, be

useful and introduces 3D-graphs of solutions. We will pass on

this. For simplicity, we will assume that first partial derivatives for all

x, y.

P(x, y), Q(x, y) are continuous together with their

The very important point is that given a solution


93

(x(t) , y(t)) of (*) , then as t varies it describes parametrically a curve in the x, y-plane. This curve is called a trajectory or orbit for the following simple reason which is an easy consequence of the chain rule: If (x(t) , y(t) ) is a solution of (*) and c is any real constant then (x(t+c) , y(t+c)) is also a solution.

But if (x(t) , y(t)) describes a trajectory then (x(t + c) , y(t + c) ) describes the same parametric curve, but with the time shifted, and hence describes the same trajectory. Quick, an example, Herr Professor!

x = y, iJ

-4x (corresponding to x + 4x = 0). If x(O) = 1 , y(O) = 0 then x(t) = cos 2t y(t) = -2 sin 2t so x(t) 2 + 1 . The trajectory is an ellipse traveled counterclockwise and we have the phase plane picture

�

=

,

2

=

y

X

-I

-2

This trajectory represents every solution that satisfies the initial conditions 2 x(to) = xo. y(to) = Yo. where x� + l'f 1 . For instance, x(t) = cos 2 ( t - 11'/2 ) , y(t) = -2 sin 2(t - 11'/2) lies on the ellipse and satisfies the initial conditions x (�) = 1, y(11'/2) = 0, or the initial conditions x (i) = t. y (i) = J3. =

Remark. To determine the direction of travel it is usually easy to let x > 0 and y > 0 and determine the direction of growth. In the example above if y > 0 then x > 0 so the x-coordinate increases, and if x > 0 then if < 0 so the y-coordinate decreases. The next important point is that trajectories don't cross. If they did you could reparametrize by shifting time, and create the situation where two distinct solutions passed through the same point at the same time. This would violate uniqueness of solutions of the IVP. Our phase plane is now filled with nonintersecting trajectories waltzing across it "filled" is right since for every point (xo , Yo) and any time to w e can find a solution of (*) with x(to) xo, y(to) xo . What else could it contain? Points (xo, Yo ) where P(xo, y0) = Q(xo, yo) = 0 called equilibrium points, or critical points. If we think of the system (*) as describing motion then an equilibrium point would be a point where x = if = 0 so the system is at rest. =

=

94

Ordinary DIHerential Equations-A Brief Eclectic Tou r

What else? It could contain closed curves whose trajectories represent periodic solutions. For such a solution it must be the case that ( x( t + T), y( t + T)) ( x( t) , y( t)) for all t, and some minimal T > 0 called its period. Note that since the equations are autonomous, the period T is unspecified so it must be calculated or estimated from the differential equation. This is often a formidable task when P and Q are nonlinear. Closed trajectories are called cycles and a major part of the research done in differential equations in the last century was the search for isolated ones or limit cycles. By isolated is meant there is some narrow band enclosing the closed trajectory which contains no other closed trajectory. An informal definition would be that for a limit cycle, nearby solutions could spiral towards it (a stable limit cycle), or spiral away from it (an unstable limit cycle), or both (a semistable limit cycle). =

stable

semistable

unstable

The important point to be made about the phase plane is that the nonintersecting trajectories, possibly equilibria, cycles, and possibly limit cycles, are all its possible inhabitants. This is the result of some fairly deep analysis and that the universe being studied is the xy-plane. It has the special property that a closed, simple curve (it doesn' t overlap itself) divides the plane into an "outside" and "inside", and this restricts the kind of behavior we can expect from solutions of (*) · Nevertheless, the phase plane can be spectacular in its simplicity.

3

The Linear System and the Phase Plane

For the 2-dimensional linear system :i;

=

ax + by,

iJ

=

ex + dy ,

clearly (0,0) is an equilibrium point and to ensure it is the only one we stipulate that ad be # 0. The case where ad be 0 will be mentioned briefly later. A discussion of this usually follows some analysis of linear systems, so expositors are eager to use all the stuff they know about � A: where A = ( � � ) . This often confuses matters and it is much easier to use the second order equation,

-

-

=

=

x

+px+qx

=

o,

A=

(�q �p) '

since we know all about its solutions and we can describe every possible phase plane configuration for the general linear system, except one. We analyze the various cases of roots >.1 , >. 2 of the characteristic polynomial (>. = >. 2 + >. + and the important

p)

p q,


95

point is that the pictures for the full linear system will be qualitatively the same, modulo a rotation or dilation.

1.

->.� , ->.2

negative and distinct.

x(t) x(t)

c2

=

= =

c1 e- A't + c2 e -A 2 t y(t) - A1 Ct€- A 1t - A2 C2 e- A2 t =

0 : (x(t) , y(t)) -+ {0, 0)

as

0 : (x(t), y(t) ) -+ {0, 0)

as

straight line going into the origin.

c1

=

straight line going into the origin.

Now suppose

>.2 > >.1

t -+

oo,

and

y(t)

=

->. 1 x(t) ;

the trajectory is a

t -+

oo,

and

y(t)

=

->. 2 x(t) ;

the trajectory is a

then

so all trajectories are asymptotic to the line origin is a

y

=

->.1 x and we get this picture-the

stable node which is asymptotically stable. y

X

y =- A. 1 x

Note that the straight line trajectories do intersect at the origin when not contradict the statement that trajectories do not intersect.

2.

>.1 , >.2

distinct and positive.

>.� , >.2

complex conj ugate with negative real part, so

reversed-the origin is an

3.

s o clearly

= =

= oo

which does

The previous picture is the same but the arro ws are

unstable node.

>.1 , 2

=

phase-amplitude form of the solution:

x(t) x(t)

t

r ± io:, r < 0 .

Use the

Ae-rt cos(o:t - ¢) , y(t) -r Ae-rt cos(o:t - ¢) - o: Ae - rt sin {o:t - ¢ ), =

(x(t) , y(t)) -+ {0, 0)

as

t -+

y(t) + rx(t) 0:

=

oo .

Now do a little manipulation:

- Ae-rt sin(o:t - ¢)

and therefore the traj ectory is represented by the equation

( u �x ) 2 + x 2

=

A 2 e - 2rt ,

96


and expanding

In

times past, when analytic geometry was not a brief appendix in a calculus text,

one would know that a rotation of axes would convert the left-hand side to an elliptical form. Since the right-hand side is a positive "radius" approaching zero as

t ---+

oo-the

origin is a stable spiral which is asymptotically stable. y

X

4. .>.1 , >.2 complex conjugate with positive real part. The previous picture is the same but the arrows are reversed-the origin is an unstable spiral.

5. - .>.1

c2

c1

= =

< 0 < >. 2 . x( t)

=

Ct e - .X l t + C2 e .X 2 t

x(t)

=

y (t)

=

- .>. l cl e - A l t + A2 C2 e .X 2 t

0 : (x(t) , y (t)) ---+ (0, 0) as t ---+ oo and y (t) - >.1 x(t) 0 : (x(t) , y (t)) ---+ (0, 0) as t ---+ - oo and y (t) >.2 x(t) =

=

We get two straight line traj ectories, one entering the origin and the other leaving and the remaining . trajectories approach them as saddle point.

t ---+ ±oo---the origin

X

6. We have previously discussed the case >.

=

±io:.

x(t) = A cos(at - ) x(t) = y(t) = -o:A sin(o:t - cp)

}

is an unstable


97

The trajectories are families of ellipses-the origin is a

center which is stable, but

not asymptotically stable. Note that the trajectories are cycles since they are closed curves, but not isolated ones so they are not limit cycles. y

X

7.

-A

< 0, a double root.

x(t) = c 1 e - .Xt + c2 te - .Xt x(t) = y(t) = -ACt e-At - AC2 te-A t + C2 e-At = -Ax(t) Then

(x(t), y(t))

�

(0, 0) as

entering the origin. If is a

c2

t

�

oo, and if

+ C2 e -At

c2 = 0 the trajectory is

a straight line

=f. 0 all trajectories are asymptotic to that line-the origin

stable node which is asymptotically stable. y

X

8.

A > 0, a double root.

The previous picture is the same but the arrows are reversed

the origin is an unstable node.

= 0,

so

Ct - ±y(t)

or

There is a degenerate case corresponding to the second order equation x

the eigenvalues are

a > 0. -ax(t) + ac1

and suppose

y(t)

=

A

=

0,

The

A

=

-a.

Then

(x(t) , y(t))

�

(ct . 0)

as

t

�

oo and

a family of straight lines. The picture is

x(t)

=

+ ax

98


y

X

and it implies every point on the x-axis is a critical point. If a < 0 the arrows are reversed. There are several other degenerate cases corresponding to the case where det A 0, but the condition det A =f. 0 assures us that (0,0) is the only equilibrium point. Using the second order equation to model the various cases does not pick up one case found in the full linear system. It corresponds to the system =

so y ( t ) = � x ( t ) , and if q < 0, then ( x ( t ) , y ( t )) ---> (0 , 0). The trajectories are straight lines of arbitrary slope going into the origin; if q > 0 they leave the origin. These are also called nodes, or more specifically, proper nodes, and the origin is asymptotically stable if q < 0 and unstable if q > 0.

X

r 0, g(O, 0) > 0, and therefore (0, 0 ) is an unstable node. The growth of X in the absence of Y (y 0) is governed by the equation x x f ( x, 0 ), so there could be the possibility of a stable equilibrium K (carrying capacity). This would imply that f(K,O) = 0 and Kf'(K,O) Kf (K,O) < 0. Note: there doesn't have be such a K e.g., x ax - bxy, a, b positivex and if y 0 then x = ax which is (0, 0)

We have u

:

v

=

=

=

=

=

=

to

=

=

exponential growth.

(K, 0)

:

Then

( < 0) 0( Kfx (K,O) Kfy(K,O) ) . 0 g(K,O) It could be that g(K, 0) 0 in which case (K, 0) would be a degenerate stable node, but we will ignore that possibility. Then we expect that if (K, 0) were stable then competition A=

=

+


would favor

1 01

X whereas Y would perish, so g(K, 0)

0 ::} (K, 0) a saddle,

Here J plays a similar role as K for X, and the analysis is the same with g(O, J) replacing f(K, 0) . We have made considerable inroads on what the phase plane portrait will look like with minimal assumptions on the nature of f(x, y) and g(x, y). At this point the various possibilities are:

(0, J)

:

y J

(i)

-

0

y

� K

J

X

(ii)

......

"

0

K

y

y J

X

0

-

(iii)

�

K

J

X

(iv)

'\ """

0

K

X

Now, the question becomes whether there exists one or more equilibrium points = g(xoo , Yoo) = 0 . If such a point were stable we have the case of coexistence; if it were unstable we have the case of competitive exclusion. In a very complex model there could be both (this might also occur in a simple model if some exterior factor like harvesting were imposed), so that for instance when X is small enough both survive, but if it is too big it wins. The case where there is one equilibrium (x 00 , Yoo ) can now be analyzed given specific forms for f(x, y) and g(x, y) . The usual textbook models assume that the growth of each population in the absence of the other is logistic:

(xoo, Yoo). Xoo > 0, Yoo > 0, where f(xoo , Yoo)

(

± = rx l -

; ) - xo:(x, y) ,

( }) - y/3(x, y)

y = sy l -

where r, s, K, and J are all positive, and o:(x, y), /3(x, y) are positive functions for x > 0, y > 0. We will stop here with the competition case, but note that the previous graphs already say a lot. In cases (i) and (ii) there need not be a point (x 00 , Yoo) so in (i) Y wins no matter what, and in (ii) X wins. For graph (iii) we suspect that there would be a stable equilibrium (x 00 , Yoo). whereas in (iv) it would be an unstable one. Predator-Prey Models

Recall that the model is

±

= xf(x, y), y = yg(x, y)

and the assumption that the per capita rate of growth of the prey X decreases as the predator Y increases, whereas the per capita growth of Y increases as X increases.

1 02


Therefore

af {)y (x, y)

0,

X > O, y >

A is

and the matrix

( < 0) + xfy (x, y) _ f(x, y) xfx (x, y) Ag(x, y) + ygy(x, y) Y9x (x, y)

(

(>

0)_ be that X

Further natural assumptions would the absence of predators, and if extinct when

0

x

=

X

increases if

y

=

)

.

(xoo ,Yoo )

0, hence the prey grows in

is the sole or major food supply of

Y,

then

Y

goes

0.

(0,0) must be a saddle. If the prey population has a growth law that leads to a stable carrying capacity K, we do not expect the predator population to grow extinct near K since it has lots of little creatures to munch upon 1 • This suggests that an equilibrium point (K, 0) is another saddle. These assumptions mean that the equilibrium point

Given the above, we can already construct the phase plane portraits y

y

Q

L..___.._ _; _____�

or

X

and now all that remains is to analyze the stability of any other equilibria. The ever popular model is the Lotka-Volterra model whose equations are

x

=

rx - axy, iJ

where all parameters are positive. Then

Y

goes extinct in the absence of

X.

X

=

-sy + f3xy

grows exponentially in the absence of

The equilibrium point

( s I f3, rIa)

Y and

is a center for

the linearized model and it is preserved in the full model. This leads to the wonderful harmonious universe phase plane portrait of ovoid shaped periodic solutions filling the region

x > 0, y > 0.

The author has always been bothered by this model, picturing a situation where there are two rabbits quivering with fear and more tragic scenario would

10n coyotes-not to worry things will improve.

be if both rabbits

A

were of the same sex. But setting flippancy

aside, the Lotka-Volterra model lacks a requirement of a good model-structura l stability. This can

be

loosely interpreted as robustness of the model under small perturbations.

For instance, modify the equation for

X

assuming that in the absence of

is logistic with a very large carrying capacity

1

K

=

llf., f. >

0 and small.

Y its growth One obtains

However, an amusing possibility would be that if the prey population gets big enough it starts destroying

predators (the killer rabbits model). Then ( K, 0) would be a stable node.

1 03


the equation for

X

rx(1 - �:x ) - axy rx - axy - �:rx2 , which is certainly a small perturbation of the original equation. The new equilibrium point (x00, Yoo) will be (sf /3, r/a - �:sf f3a) which will be as close as we want to (sf/3, r/a) x= =

if f is small enough. But a little analysis shows that it will be a stable spiral or a stable node for that range of f so the periodicity is lost. An interesting discussion of further variants of the Lotlca-Volterra model can be found in the book by Polking, Boggess, and Arnold.

5

Harvesting

If we consider the standard competition model with logistic growth for both populations,

and where their per capita rate of growth is reduced proportionally to the number of competitors present, we have the model

x

=

=

iJ

=

=

rx ( 1 - ; ) - axy x(r - rxjK - ay), sy ( 1 - �) - f3xy y(s - syjJ - f3x),

where all parameters are positive. The linear curves in parentheses, called nullclines, graphed and we get the possible pictures J

are

y

ria

or

0

X

K

0

s/ 13

X

Competitive Exclusion

Coexistence

where the equilibria are circled. Now suppose the population Y is harvested at a constant rate H > fishing, or disease, for instance. The second equation becomes iJ

=

0 caused by hunting,

y(s - syjJ - /3x) - H

and the right-hand side is a hyperbolic curve whose maximum moves downward as H is increased. The successive pictures in the coexistence case look like

1 04


y

y

y

r/a

r/a

0

K

K

and note that the equilibria (0,0) and (K, 0) have moved below the x-axis. Along and near the x-axis we have !fit � - H < 0 so that once there are no equilibria in the positive xy-quadrant, then X wins all the marbles (or birdseed). The model is quite similar to that for the one-population logistic model with harvesting discussed in Chapter 2. The critical value He of the harvest rate is easily found by solving

rx r - - - ay = O K

for x, then substituting it into y(s - sy j J - f3x)

-H=0

and solving the resulting quadratic equation. The value of H beyond which there will not be real roots will be He. Example.

A system with coexistence is X

=

X

[� � � ] -

X-

4

y 1 3

y

'

y

=

[� � -

5 0

y-

�]

X ' 1 3

with saddle points (200,0) and (0,400), and (x0, , Yoo ) = (50, 375) a stable node. To determine the critical harvest H = He when the y-population is harvested, first solve for x:

1 1 1 - - -x - -3 y 2 400 10 Next solve y

[� - --

-11 y- 3 500 10

5

which simplifies to 2y2 - 750y + 1250H y

=

=

(

=

0

::::}

200 -

x

2 5

= 200 - - y.

�5 y

)]

-H=0

0. Its roots are

� (750 ± V7502 - 1Q4 H )

and they will no longer be real when

104 H

>

7502

or

H

>

7502 /104

=

56.25

=

He.

X

1 05


One can apply harvesting to competitive exclusion cases and create coexistence, as well as to predator-prey models to destabilize them. More complicated models than the logistic type ones could make interesting proj ects, possibly leading one to drop mathematics altogether and take up wildlife management. Happy Harvesting !

6

A Conservative Deto ur

A pleasant respite from the onerous task o f solving nonlinear equations in several variables to find equilibria, then linearizing, calculating eigenvalues, etc . , is afforded by examining

conservative systems.

No knowledge of political science is required, and the analysis

of the nature of equilibria in the two-dimensional case is simply a matter of graphing a function of one variable and locating its maxima, minima, or inflection points-a strictly calculus exercise. Since the differential equations model linear or nonlinear, undamped, mass-spring systems or pendulums, one might consider briefly studying them prior to embarking on the discussion of almost linear systems. The standard equation for a one degree of freedom conservative system is

x + f(x) where

x

=

x(t)

is a scalar function. This could describe the motion o f a particle along

a line or curve, where the restoring force so there is

i;

i.e. no

no damping,

the undamped oscillator,

f ( x)

=

spring. We will assume that

f(x)

f(x)

f sin x, the pendulum,

or

f ( x)

=

f(x) = k 2 x, x + x3, a nonlinear

is sufficiently nice so that its antiderivative

i;

=

f(x).

x± + f(x) ±

then integrate t o obtain the

is only dependent on the displacement,

term present. Some simple examples are

r f(s)ds is defined and satisfies F' (x)

by

0,

=

=

F(x)

=

Now multiply the differential equation

o,

energy equation:

j; 2

'2

+

F(x)

=

C,

a constant.

This is the key equation; the first term on the left is a measure of the kinetic energy per unit mass, and the second is a measure of the potential energy per unit mass. Consequently, the energy equation says that along a solution (kinetic energy) + (potential energy)

=

constant,

which is the defining characteristic of conservative systems-their total energy is constant along trajectories. Moving to the phase plane, where

E(x, y) and we can define

z

=

E(x, y)

(x, x) 1

=

as the

'2 y

=

(x, y),

2 + F(x)

the energy equation becomes

=

energy surface.

C, Its level curves

E(x, y) = C,

in

1 06

Ordinary DIHerentlal Equations-A Brief Eclectic Tour

the phase plane, are therefore the trajectories of the two-dimensional system i:

=

y, iJ = - f(x) .

For the simple oscillator f(x) = x, so F(x) = ! x 2 , and the energy surface is the paraboloid z = !Y2 + !x 2 whose level curves are circles !Y 2 + !x 2 = C, reflecting the fact that the equilibrium point (0,0) is a center. The diagram below shows the values of the total energy C = Kt , K2 . z

- - -'- - '

- ....._

I

"

\

y

X

The equilibria of a conservative system are points (x, 0) where f(x) = 0, and since F' (x) = f(x) they are critical points of F(x); this is the key to the analysis. Rewrite the energy equation as 1 2 2 y = C - F(x) or y = ± V'i Jc - F(x) ,

and then compute the values of y near x = x for selected values of the total energy C. For each x and C there will be two such values, in view of the ± sign, which tells us that trajectories will be symmetric about the x-axis. For gorgeous, exact graphs this plotting can be done using graphing technology, but to sketch trajectories and analyze the stability of (x, 0), we only need eyeball estimates of the quantity JC - F(x) near the critical point x. The various possibilities are: (i) F(x) is a local maximum. Plot z F(x) near x = x then select values z = K1 , K2 , , of the total energy and intersect the horizontal lines z = Ki with the graph. Align an (x, y) graph below the (x, z) graph then plot or estimate the points (x, ±V2 JKi - F(x) ) . We see that (x, 0) is a saddle point. (ii) F(x) is a local minimum. This case is sometimes referred to as a potential well or sink. Proceed as in i) and we see that (x, 0) is a center. =

•

.

•

1 07


----

rn

- -

---

- - - - - - -

_ _ _ _ _ _

' '

-

-

- -

-

- - - - -

-

- - - - - -

i3

_ _ _ _ _ _

2_ KI _

-

-

-

-

- - - - - - - - - - - - - - - - - -

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

' X

X 0

-

-

-

-

- - - - - - - - - - -

X

y

X

i)

ii)

(iii) (x, F(x) ) is a point of inflection. The trajectory through (x, 0) will have a cusp and it is sometimes called a degenerate saddle point-it is unstable.

X

1 08


Since i), ii) and iii) are the only possibilities we can conclude that conservative systems x + f(x) = 0 can only have equilibria which are saddle points (unstable) or centers (stable), and there will be no asymptotically stable equilibria. To plot a specific trajectory, given initial conditions

(x( to ) , x( to ) )

(x( to ) , y( to ) )

=

=

(xo , Yo ) ,

use the fact that its total energy i s constant. Hence

is the required implicit representation. Examples. We start off with the classic

f sin

x = 0, the undamped pendulum. Then f ( x) = f sin x so F( x) = which has local minima at x ±2mr and local maxima at x ± (2n + 1)11", n = 0, 1 , 2, . . . Given total energy C, trajectories will satisfy the equation

1.

x +

- f cos x

=

=

.

y and the picture is:

=

J

± v0. c +

f cos x,

z

I I Yl I I I

This is a local picture which shows that a moderate perturbation from rest (x y = 0) will result in a stable, periodic orbit, whereas a small perturbation from x ±1r, y = 0, the pendulum balanced on end, results in instability. A more complete picture of the phase plane looks like this =

=


1 09 y

X

The unbounded trajectories at the top and bottom of the graph represent trajectories whose energy is so large that the pendulum never comes to rest. For 0 < C < f there is a potential well centered at (0, 0 ) , which implies that initial conditions l x (O) I < 1r,

1 ± ( 0 ) 1 = l y(O) I
.x 2 , 0 < >. < 1/4, but examine the case t - 1 if x < - 1 c) f(x) = 0 if l x l < 1 1 if x 2: 1 . a)

=

{

�

>.,

An important question i s the following one: given a closed orbit of a nonlinear system, representing a periodic solution, what is its period T? For most autonomous systems this is a difficult analytic/numeric question. If one is able to obtain a point (xo , Yo) on the orbit, then one can numerically chase around the orbit and hope to return close to the initial point, and then estimate the time it took. If the orbit is an isolated limit cycle even finding an ( xo , Yo ) may be a formidable task. But for two-dimensional conservative systems it is easy to find an expression for the period. We will assume for simplicity that the orbit is symmetric with respect to the y-axis so it looks like this: y

X

-a

+ f(x) = 0, then when x = a, x = y 0 so the energy equation is ·2 E(x, y) = E(x, x) = + F(x) = F(a), F'(x) f(x),

If the equation is x

=

�

=

112


or

�;

=

J2

(F(a) - F(x)) 1 1 2 .

We have used the + sign since we will compute in the region x equation implies that 1

= y

� 0. The last

dt = J2 dx (F(a) - F(x)) l / 2 ' and if we integrate along the or�it from x 0 to x a we get 114 of the time it takes =

=

to traverse the orbit, hence

Period = T(a)

= =

Jro J2 (F(a) F(x) ) l /2 dx 1 2J2 r oJ (F(a) - F(x) ) l l 2 dx. 1

4

_

The expression for the period is a lovely one, but has a slightly venomous bite since the integral is improper because the denominator vanishes at x = a. Nevertheless, the period is finite so the integral must be finite-with care it can be evaluated numerically. Let' s look at the pendulum equation g . x.. + sm x = 0 , x(O) = a, ± (o ) = o,

L

and some approximations:

sin x x and we' ll play dumb and suppose we don't know how to solve x + fx = 0. Then F(x) = f z22 and

1st Approximation:

T(a)

�

= 2J2 r

Jo

1

1 dx y!( ;2 - z; ) 1 2

=

4

which is independent of a and valid only for small then T � 2.007090. 2nd Approximation:

sin x

�

x - �3 , then F(x)

T(a) = 2J2 We computed it for g = 9.8 mj sec 2 .

{o a

Jo Vf(� 2

_

=

�4

1 a = 21r {f V{f9 sin - ::_] a o V9

l x l - If L

f ( z22 _

=

1m. and

= 9.8 m/sec 2

�: ) and

� dx.

) 2 + 24

�

g

2 a = 1/2, 1 , 3/2; the second column is the value for L T

(�)

= 2J2

T( 1 )

= 2 J2

T(3/2)

= 2J2

/% /% /%

(2.257036)

2.039250

(2.375832)

2.1 46583

(1.628707)

2.375058

=

1m. and

5

113

Linear and Nonlinear Systems

3rd Approximation:

T(a) Then

as

=

above

T

(�)

-f

cosx and 1 2v'2 1a .Jf(cosx - cosa) 112 dx.

the full pendulum so F ( x )

=

o

=

2J2

= 2J2 T(3/2) = 2J2 T(1)

/¥ /¥ /¥

(2.256657)

2.038907

(2.368799)

2.140229

(2.584125)

2.334777

For a large class of conservative systems there is a direct relation of solutions and periods to Jacobi elliptic functions and elliptic integrals. The article by Kenneth Meyer gives a nice discussion of this. If the reader is in a number crunching mood he or she might want to look at: a) f ( x )

=

x 3 , with x ( O )

= a, ±(0) = 0; a = 1, a = 24.

b) The step function f ( x ) given in c) above and determine the minimum period of oscillation.

= x2

c) The example f ( x ) x + given previously; the limits of integration in the period integral must be adjusted since the orbit is not symmetric in the x-direction. d) Compare the approximate and full pendulum's period when 31/10, so the pendulum is nearly vertical.

a=

a is close to

n,

say

Finally, we want to briefly look at the effect of damping on a conservative system. When we examined the second order linear equations

we saw that the addition of the damping term c± resulted in all solutions approaching zero as t --+ oo. In the phase plane the origin, which was a stable center in the first equation, became an asymptotically stable spiral point or node. A simple phase plane analysis for the damped pendulum x + c± +

f sin x = 0,

0 < c < 2V9jL,

shows that the saddle points of the undamped pendulum are sustained, whereas the centers become asymptotically stable spiral points. For the conservative system x + f( x) 0 we can model the addition of damping by considering the general equation

=

x + g ( x, ± ) ± + f ( x )

= 0,

114

Ord inary Differential Equations-A Brief Eclectic Tour

where the middle term represents the effect of damping-g( x, x) could be constant as in the case above. Multiply the equation by x and transpose terms to obtain xx

and if F'(x) =

+ f(x)x

=

-g(x, x)x 2 ,

f(x) this can be written as 2 + F(x) = -g(x, x)x2 •

:t ( �

)

But the expression in parentheses is ·the energy equation, and if we let obtain the equation

x

=

y then we

! E(x, y) = -g(x, y)y2

which gives us a measure of the rate of growth of the energy of the system. For instance, if g(x, y) > 0 for all x, y, the case of positive damping, e.g. g(x, y) = > c 0, then the energy of the system is strictly decreasing, so any nontrivial motion eventually ceases-the system is dissipative. For any conservative system x + f(x) 0, results in the energy growth equation

Example:

ex,

c

>

d E(x, y) = dt

=

- cy

0 the addition of a damping term

2

so all motions eventually damp out. Similarly for the equation x + have

lxlx + f(x)

=

0, we

and the behavior is the same. On the other hand if g(x, y) < 0 for all x, y, the case of negative damping, then the energy of the system is increasing. The effect is that of an internal source pumping in energy. But the most interesting case is one where both kinds of damping occur in different regions of the phase space, and the balance between energy loss and gain can result in self-sustained oscillations or limit cycles. A simple example is the following one:

and the energy growth equation is

When x 2 + y2 < 1 the damping is negative, so energy increases, whereas when x2 + y 2 > 1, the damping is positive and energy decreases. It is easily seen that x( t ) = cos t is a solution, which gives an isolated closed trajectory x2 + y 2 = 1 in the phase plane, hence a limit cycle.

5

1 15


y

X

The extensively studied Van der Pol equation

x + E (x 2

- 1 ) ± + x = o, E

> o,

has the energy growth equation

lxl < 1

For

energy is increasing, whereas for

expect to find self-sustained oscillations.

lx l

>

1

it is decreasing, so we would

It has been shown that for any

exists a stable limit cycle; the proof is complicated but elegant. For approximated by for

E

�

1

x (t)

= 2 cos t,

E

hence is circular in shape, with period

< 0. 1

E

>

0

there

it is closely

T � 211",

whereas

it becomes quite jerky and is an example of a relaxation oscillation. X

0

Enough !

The author' s detour has arrived at a point where it is best to return to the

main highway, described.

and leave the reader to further explore the fascinating territory just briefly

116 7

OrdlnaiJ Differential Equations-A Brief Eclectic Tour

Sta b ility and Gronwa l l-ing

Up to this point, except for several nonautonomous frrst order equations in Chapter 2, the discussion of stability of solutions has been limited to an analysis of the solution x(t) = 0 of constant coefficient second order equations, and the equilibrium point (0,0) of the system � = A : , where A is a 2 x 2 constant matrix. We want to move a little further afield and to do so we need a measure of distance or a norm in First of all if : = col (x 1 , is a point (vector) in then define the norm of : , by Preliminaries

.

•

.

Rn .

Rn

Xn )

n

This is not the Euclidean norm, but is equivalent, is easier to work with, and has all the required properties. Then we need a measure of the size of a matrix, so if A = ( aii ) , i, j 1 , . . , n, is an n x n matrix we define the norm of A by =

.

II A I I

n

=

L l aii l ,

i ,j=l

and it has the properties we need:

II A + B ll ::; I I A I I + I I B I I , I I AB II ::; I I A I I I I B II , llcAII ::; lc i i i A II for any scalar c, II A : II ::; I I A II II : II for any vector : · Obviously, if A = A(t) is continuous, i.e. has continuous entries aij (t), then I I A(t) ll

is continuous. To precisely define the notion of stability and asymptotic stability of a solution : (t), defined for t0 ::; t < oo , would require some delta-epsilonics, which the reader can find in any intermediate or advanced text. The following picture describes stability of the solution : (t) = : o ; we have selected a constant solution for artistic simplicity. X

x(t) �(t)

=

�0

Stability What the picture means is that given an arbitrary € > 0 we can construct a "tube" of radius € around :(t) = : o. and any solution : (t) satisfying : (to) = : 1 · where : 1 is in a smaller shaded disc centered at (t0, :o), will be defined and stay in the tube for all t 2': to. Once close-it stays close.

5

117


The above definition does not prevent a solution from entering the tube at a later time than t0 then leaving it. To preclude this we need the much stronger notion of uniform stability which we will not discuss. For asymptotic stability of the solution � (t) = � o the picture is this one: X

t 0

Asymptotic Stability

We can say that � (t) once close-it stays close, and furthermore the diameter of the tube goes to zero as t --+ oo, so lim x (t) = xo. t�OC> I".J

I"'V

The reader's mind's eye can now substitute a nonconstant solution � (t) satisfying � (to ) = x0, and construct a wiggly tube around it to infer the notion of what is meant by stability and asymptotic stability of the solution � (t).

StabiUty for Linear Systems

For the linear systems � = A � , A = (aij ) . i, j = 1 , . . . , n, or x = A(t) � , A(t) (aij (t) ) , i, j 1 , . . . , n, where each aij (t) is continuous for t 2:: to. we know the solution � (t) of the NP =

=

x = Ax or

� = A(t) � ,

is given via the fundamental matrix <Jl (t) � (t )

=

=

� (to ) = �o ,

('Pij (t) ) ,

<Jl (t ) � o ,

<Jl (to) = I ,

and the columns of <Jl (t ) are a fundamental system of solutions. From the last statement we can imply several things about l l 0, a > 0 such that I ! <J� (t) l ! :::;

R

Re-o. t

118


(

. . e-3t e-3t +t e-3t A = ( 1 -1 _ 2 ) , a fundamental matnx lS -e-3t - - t e-3t ( �1 =i ) to get � (t) satisfying � (0) = I , Multiply by O(o) - 1 Example.

_4

=

)

f"\ u

=

(t) .

Then

A little analysis shows that if 0 � then

a

< 2.63 then e - a t > te-3t for t > 0, so let

a =

1

The first case of stability to logically consider is the general one A = A(t), but we must know the structure of �(t) which will usually be very difficult to find. Here's a sometimes helpful result: If all solutions of !

=

A(t) �;. t 2: t0, are bounded then they are stable.

� 1 (t) be the solution satisfying � 1 (to) = � O· and � 2 (t) be the solution satisfying x 2 (to) = ""x 1 . Since all solutions are bounded we know l l � (t) l l M, so let l l xo - x 1 ll < "" E/M and the simple estimate

Let

,.....,

�

f",J

gives the desired conclusion. For a constant matrix

A things

get better:

If all the eigenvalues of A have negative real parts then every solution of ! is asymptotically stable. Just replace the constant M in the previous estimate with

This shows that solutions are bounded since

-+OCl (x 2 (t) - ""x 1 (t) ) = 0.

lim

t

'V

ll � (t) ll

Re-o:t, R > 0,

�

Re-o:t ll �o l l

a

>

=

A�

0 to obtain

and furthermore

I"V

Way back in Chapter 2 we introduced Gronwall ' s Lemma and touted it as a very useful tool to study stability. To live up to our claim we examine the nonautonomous system

! = (A + C (t)) � , � (0)

=

£ ; C (t)

=

(cii (t)) ,

i, j

=

1, . . .

, n,

where each Cij (t) is continuous for 0 � t < oo. Our intuition would tell us that if C (t) were "small" in some sense, the behavior of solutions would mimic those of the system ! = A� . We follow that line of reasoning. First, recall that for the general IVP

! = A� + � (t) , � (0) = � o

5

119


the solution is given by the variation of parameters formula

� ( t ) = � (t) � o +

1t �(t - s)J!(s)ds.

The trick is to rewrite the system � = (A + C(t)) � as � = A � + C(t) � and let the last term be our J!(t). Then we can express its solution as the integral equation

� (t) = �(t) � o +

1t � (t - s)C(s) � (s)ds

which is ripe for Gronwall-ing. Suppose all the eigenvalues of A have negative real parts, then ll� (t) ll $ some positive a and R and t 2: 0. Take norms in the previous equation

ll � (t) ll

$ $

Re- cxt, for

1t ll �(t - s) II I I C(s) l l ll � (s) ll ds t Re - cxt l l � o ll + 1 Re -cx (t - s ) II C(s) ll ll � (s) i i ds, ll � (t ) ll ll � o l l +

and multiplying through by

ea t gives

l l � (t) ll e cxt

$

Rll � o ll +

Now apply Gronwall' s Lemma

1t RII C(s) il ll � (s) il ecxs ds.

[ 1t II C(s) ll ds] 00 Rll � o ll exp [R 1 II C(s) ll ds] , 00 Rll �o l l e- cxt exp [R 1 IIC (s) ll ds] .

ll � (t) ll ecx t

$

Rll � o ll exp R

$

or

ll � (t) ll

$

We conclude that if the integral

f000 II C(s) li ds is finite then

i) All solutions are bounded, hence stable, ii) All solutions approach zero

as

t --+ oo since

a

>

0.

But the system is linear and homogeneous, so the difference of any two solutions is a solution. We can conclude

� = (A + C(t)) � . where C(t)

is a continuous matrix for 0 $ t < oo. If the eigenvalues of A have negative real parts and f000 II C(t) i idt < oo, then all solutions are asymptotically stable. Given

Example. x

(

+ 2 + 1� 2 t

)

x

+ x = 0 can be expressed as the system

120


The matrix A has a double eigenvalue >.. = - 1 ,

and

100 0

1 + t2

--

1

dt

is finite so all solutions are asymptotically stable. The conditions on the growth of C(t) can be weakened, but the intent was to demonstrate a simple application of Gronwall' s Lemma, not give the definitive result. Stability for Nonlinear Systems

At this point in the development of potential ODE practitioners, their only likely exposure to nonlinear systems are to those sometimes referred to as almost linear systems. They may come in different guises such as conservative or Hamiltonian systems, but the basic structure is the following one, where we stick with n = 2 for simplicity: Given the autonomous system

x

=

P(x, y),

y

Q(x, y),

=

where P and Q are smooth functions, and (0,0) i s an isolated equilibrium point. Then since P(O, 0) = Q(O, 0) = 0 we can apply Taylor' s Theorem and write j;

=

Px (O, O)x + Py ( O, O)y +

(L)

Y - Q x (O ' O) x + Q Y (O ' O) y + .

_

( (

�

hig er order terms m x and y

) )

higher order terms . . m x and y

The higher order terms will be second degree or higher polynomials in x and y. H the equilibrium point is (x0 , y0 ) -1- (0, 0) we can make the change of variables

x

= u

+ xo ,

y

= v

+ Yo ,

and this will convert the system (L) to one in u and v with (0, 0) an equilibrium point; the first partial derivatives will be evaluated at (xo , Yo ) . The system (L), neglecting the higher order terms, is the linearized system corresponding to the original system, and we use it to study the stability of (0,0). H we let �

=

col(x1 , . . .

, Xn

) we can write a more general form of (L):

where A is a constant n x n matrix and / ( � ) is higher order terms, which may be infinite in number, as in the Taylor series expansion. What is important is that /( � ) satisfies

md this relation is independent of the choice of norm. Systems of this form are called 1lmost linear systems.

1 21

5 Linear and Nonli near Systems Examples.

a) Suppose the higher order tenns (H.O.T. if you want to appear hip) are x2 + 2xy + y4 . Since 1 1/ lx l x l l + IYI $ 1/ IYI

{

then

II ! ( :) I I --

11: 11

l x2 + 2xy + Y4 1 < l x l 2 + 2 l x i i Y I + IYI 4 ...!. ,...-!-....:....:. .:.:;: ....:.: ..,.... :..;:.... - .!...l x l + IYI l x l + IYI

=

2 l x i iY I � lxl2 < = alxl + IYI3 + + x l l IYI IYI which approaches 0 as l x l + IYI --+ 0. b) But one could use the Euclidean norm r = Jx 2 + y2 coordinates, x = r cos (}, y = r sin (}, to obtain Il l(:) I I

=

11 : 11 , then convert to polar

l r2 cos2 (} + 2r2 cos O sin O + r 4 sin4 0 1 r

11: 11 < -

r2 + 2r2 + r4 = 3r + r3 --+ 0 as r

c) The undamped pendulum x + x. = y,

r --+ 0.

f sin x 0 can be written as the system =

g ( x - sm y. = - g sm x = -gx+ L . x) L . L

and lx - sin x l l x - (x - � + · · · ) I 3. x + lx l l l IYI

.:...,. .,..-,. ..-...,...:.
(t) is the fundamental matrix associated with the system � = A ;: . ll> (O) = I, then we can employ the same trick used in the proof of the previous stability result. If ;:(t) is the solution satisfying the I.C. ;:(O) = ;: o then it satisfies the integral equation

;:(t)

=

ll> (t);:o 4-

1t ll> (t - s)f(;:(s))ds.

The expression is ripe for Gronwall-ing but we frrst need some assumptions and estimates. Assume the eigenvalues of A all have negative real parts, so there exist positive constants R , a such that ll ll> (t) ll � Re - at , t � 0. Then

l l;: (t) l l

�

Re- a t ll;: o l l +

or

l l � (t) ll eat

�

1t Re - a (t - s ) l l f(� ( s)) ll ds

Rll � o ll +

1t Reas ll f(� (s)) llds.

Next note that the growth condition 11! ( ;: ) 1 1 = o( ll � l l ) implies that given any t > 0 (the fearsome t at last!) there exists a 8 > 0 (and its daunting side kick!) such that 1 1 ! ( � ) 11 < t ll � ll if ll x ll < 8 . Since x(t) is assumed to be close to the zero solution, we can let ll xo ll < 8 . But x (t) is continuous, so ll � (t) ll < 8 for some interval 0 � t � T, and the great Gronwall moment has arrived ! We substitute t ll x(s) ll for ll f(;:(s)) ll i n the previous estimate, Gronwall-ize, and obtain "'

"'

"'

0 � t < T, or

0 � t < T, the crucial estimate. Since t > 0 is at our disposal and so is x0, let t < a/ R and choose x0 satisfying "' ll xo ll < 8/2R . The estimate gives us that l (t) ll < 8/2 for 0 � t < T, and we can apply a continuation argument interval by interval to infer that

�

"'

"'

i) Given any solution � (t), satisfying � (0) � o where ll ;:o ll < 8/2 R, it is defined for all t � 0 and satisfies ll ;:(t) l l < 8/2. Since 8 can be made as small as desired this means the solution � (t) = 0 (or the equilibrium point £) is stable. =

tR - a < 0 this implies that lim --+oo ll x(t) ll t stable.

ii) Since

rv

=

0 hence x (t) rv

=

0 is asymptotically

We have proved a version of the cornerstone theorem, due to Perron and Poincar�. for almost linear systems:

5

1 23


� = A � + /( � )

where f is continuous for 1 1 � 11 < a, a > 0, and ) as 11 1 o( ll � ll � 1 -+ 0. If all the eigenvalues of A have negative real parts then the solution � (t) £ is asymptotically stable. Given

11/( � ) 11

=

=

Remark.

i) The result can be generalized to the case where f = f(t, � ), is assumed to be continuous for 11 � 11 < a, 0 ::; t < oo , and the estimate ll f(t, � ) II = o( ll � ll ) as 11 � 11 -+ 0 is uni�orm in t . For example, lx 2 cos t! = o ( l x l ) as l x l -+ 0, uniformly in t, since l x 2 cos t1 ::; lx l 2 .

ii) By considering the case t -+ -oo and assuming all the eigenvalues of A have positive real parts, the proof above implies that if 11/(x) ll = o( l l x ll ) as ll x ll -+ 0 then the solution � (t) = 0 is unstable.

iii) For the two-dimensional case it follows that if the origin is a stable (unstable) spiral for the linear system, it is a stable (unstable) spiral for the almost linear system, and similarly for stable (unstable) nodes. It can be shown that the saddle point configuration is preserved, whereas a center may remain a center or be transformed into a stable or unstable spiral.

The Perron/Poincar� theorem gives credence to the assertion that when we linearize around an equilibrium point �0 by using Taylor' s Theorem, the asymptotic stability or instability of the linear system is preserved. The theorem is a fitting milestone at which to end this brief, eclectic journey.

FI NALE

To those who have snarled and staggered this far, the author hopes you have been stimulated by this small tome, and will continually rethink your approach to the subject. And ALWAYS REMEMBER The subject is ORDINARY DIFFERENTIAL EQUATIONS and NOT Algebra Calculus Linear Algebra Numerical Analysis Computer Science

Refe rences

(And other tracts the author has enjoyed, with brief cornrnentarx.)

D. Acheson, From Calculus to Chaos, An Introduction to Dynamics, Oxford University Press, Oxford, 1997. A delightful book to waltz through the calculus, and the final chapters are a nice introduction to ODEs and chaotic systems.

W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 6th Ed. Wiley, New York, 1997. Old timers will remember the thin, classic grandfathers of this book, which has now grown plump like its contemporaries. But it is still an excellent introductory text.

F. Brauer and J.A. Nobel, The Qualitative Theory of Ordinary Differential Equations, An Introduction, W.A. Benjamin, 1969; (reprinted) Dover Publications, New York, 1989. One of the first textbooks on the modern theory; the discussion of Lyapunov theory is excellent.

F. Brauer and D.A. Smchez, Constant rate population harvesting: stability, Theor. Population Biology 8 ( 1 975), 1 2-30.

equilibrium and

Probably the first theoretical paper (but readable) to discuss the effects of harvesting on equations of population growth. Sandhill Cranes are featured.

M. Braun, Differential Equations and Their Applications: An Introduction to Applied Mathematics, 4th Ed. Springer-Verlag, New York, 1992. A well written introductory textbook with a solid emphasis on applications.

discussion of art forgery and Carbon- 1 4 dating is unsurpassed.

The

R.L. Burden and J.D. Faires, Numerical Analysis, 5th Ed. PWS-Kent, Boston, 1 993. An easy to read introduction to numerical methods with wide coverage and good examples.

1 27

1 28


G.F. Carrier and C.E. Pearson, Ordinary Differential Equations, Blaisdell, 1 968; (reprinted) Soc. for Industrial and Applied Mathematics, Philadelphia, 1 99 1 . Snapshot-like coverage of a lot of classic topics important in applications, with problems that are formidable.

F. Diacu, An Introduction to Differential Equations, Order and Chaos, W.H. Freeman, New York, 2000. A recent text which bucks the current trend towards obesity, and has a very modern flavor. 1 - 1 1 12 semesters of topics, like the

If you believe an introductory bOOk should cover

original Boyce and DiPrima text, look at this one.

K.O. Friedrichs, Advanced Ordinary Differential Equations, Gordon and Breach, New York, 1 966. One of the volumes of the old Courant Institute Notes which were all great contributions to modern mathematics.

W. Hurewicz, Lectures

!n Ordinary Differential Equations, MIT Press, Cambridge, 1 958.

Truly a classic ! This little paperback was one o f the first books t o introduce th e modern theory, as developed by the Soviet school, to English-speaking audiences.

D.W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, Oxford, 1 987. Lots of wonderful examples of nonlinear systems and the techniques needed to study their behavior-a very handy book to have on the shelf.

E. Kamke, Differentialgleichungen, Losungsmethoden und Losungen, Chelsea, New York, 1 948. If it can be solved, no matter how weird it looks, it' s probably in here! A more recent example of the same mad pursuit is a handbook by A.D. Polyanin and V.F. Zaitsev (CRC Press, 1 995) which has over 5000 ODEs and their solutions, including 267 Riccati equations !

W.D. Lakin and D.A. Srutchez, Topics in Ordinary Differential Equations, Prindle, Weber and Schmidt, 1 970; (reprinted) Dover Publications, New York, 1 982. Covers asymptotics, singular perturbation, nonlinear oscillations, and the relation between boundary value problems and elementary calculus of variations.

A.C. Lazer and D.A. Srutchez, Periodic equilibria under periodic harvesting, Math. Magazine 51 ( 1 984), 1 56-1 58. The authors thought this would be a nice paper for a student journal, consequently, its useful major result is almost unknown to practicing mathematical ecologists.

J.E. Leonard, The matrix exponential,

SIAM

If you really want to know how to compute

Review 38 ( 1 996), 507-5 1 2.

exp(tA)

here' s a good way to do it.

1 29

5 References

L.N. Long and H. Weiss, The velocity dependence of aerodynamic drag: a primer for mathematicians, American Mathematical Monthly 106 ( 1 999), 1 27-1 35. Imparts some wisdom into the oft stated and misused assumption that the air resistance of a falling body is proportional to the square of its velocity.

R.E. O'Malley, Jr., Thinking About Ordinary Differential Equations, Cambridge Univer sity Press, Cambridge, 1 997. A small paperback which is an excellent introduction, with fine problems, and of course, O'Malley' s favorite topic, singular perturbations, gets major billing.

R.E. O'Malley, Jr., Give your ODEs a singular perturbation ! , Jour. Math. Analysis and Applications 25 (2000) , 433-450. If you want a flavor of singular perturbations, this paper is a very nice collection of examples, starting with the simplest ones.

K.R. Meyer, Jacobi elliptic functions from a dynamical systems point of view, American Mathematical Monthly 108 (200 1 ), 729-737. The solutions and periods of conservative systems are intirnateif connected to elliptic functions and elliptic integrals . The author gives a first-rate introduction to this connec tion. .�. .._....._

V. Pliss, Nonlocal Problems of the Theory of Oscillations, Academic Press, New York, 1 966. A leading Soviet theorist gives a thorough introduction to the subject. It is an older treatment but is one of the few books that discusses polynomial equations.

J. Polking, A. Boggess, and D. Arnold, Differential Equations, Prentice Hall, Upper Saddle River, 200 I . A recent big, introductory book , with a nice interplay between computing and theory, plus a very good discussion of population models.

D.A. Sl\nchez, Ordinary Differential Equations and Stability Theory: An Introduction, W.H. Freeman, 1 968; (reprinted) Dover Publications, New York, 1 979. The author' s attempt, when he was a newly-minted PhD to write a compact introduction to the modern theory, suitable for a graduate student considering doing ODE research, or for an undergraduate honors course. Its price may help maintain its popularity.

D.A. Sl\nchez, Ordinary differential equations texts, American Mathematical Monthly 105 ( 1 998), 377-383. A somewhat acerbic review o f th e topics i n a collection o f current leviathan ODE texts. The review played a large role in the inspiration to write the current book.

D.A. Sl\nchez, R.C. Allen, Jr., and W.T. Kyner, Differential Equations, Addison-Wesley, Reading, 1988. Yes, this was probably one of the first corpulent introductory texts, now interred, but made the important point that numerical methods could be effectively intertwined with the discussion of the standard types of equations, and not as a stand-alone topic. R.I.P.

1 30


S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, Reading,

1 994.

Gives a very thorough introduction to the dynamical systems approach, with lots of good examples and problems.

S.P. Thompson, Calculus Made Easy, Martin Gardner (ed.), originally published St. Martin' s Press, New York, 1 998. This reference is here because in the 1 970' s the author wrote a letter in the

the American Mathematical Society

1 9 1 0;

Notices of

protesting the emergence of telephone directory size

calculus books . The creation of the Sylvanus P. Thompson Society, honoring the author of this delightful, incisive, brief· tome, was suggested to encourage the writing of slim

calculus texts. Perhaps a similar organization is needed today, considering the size of ODE books.

F. Verlhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag,

Berlin,

1 996.

A very well written and thorough advanced textbook which bridges the gap between a first course and modem research literature in ODEs. The mix of qualitative and quantitative theory is excellent with good problems and examples.

I n d ex A

almost linear systems, 1 20 always close, 1 5 amplification factor, 76 Annihilator Method, 72 asymptotic stability, 1 5 , 1 1 7 attractor, 14 B

beats, 77 boundary layer, 25, 5 1 Boundary Value Problems, 82 Brauer, Fred, 1 8 c

center, 97 characteristic polynomial, 64 coexistence, 1 0 1 comparison of coefficients, 7 1 , 92 comparison results, 3 1 Competition Models, 1 00 competitive exclusion, 1 01 conservative systems, 1 04 continuation, 6, 39 convolution integral, 24, 78 cycles, 94 D

damping, 68 degenerate saddle point, 1 07 dependence o!l initial conditions, 38

Euler' s Method, 44, 82 existence, 5

F

feedback-control, 27 finite escape time, 1 2 Friedrichs, K., 34 fundamental matrix, 20 fundamental pair of solutions, 58, 62

G

gain, 76 global error, 46 Gronwall' s Lemma, 38, 1 1 8

H

harmonic oscillator, 67 harvesting, 1 8 , 1 03 homespun stability criterion, 15, 84

I

Improved Euler' s Method, 44 Incredible Shrinking r, 4 1 infinite series solutions, 85 initial condition, 4 initial value problem, 5, 53 integrating factors, 21

L

E

eigenvalue problem, 83 energy equation, 1 05 energy surface, 105 equilibrium, 1 0, 14 · equilibriu i� ;;i f' "'� . . 1/tl/l._ -�

� � o :·

•

'

•

•

.

· �.

Lazer, A.C., 36 limit cycles, 94 linear equation, 20, 55 linear independence, 59 linearity, 57 linearized stability analysis, 1 5 little gnome, 45 131

Ordinary Dinerential Equations-A Brief Eclectic Tour

local error, 46 Lotka-Volterra model, 1 02 M

repeller, 14 resonance, 78, 80 Riccati equation, 26 RKF methods, 47

. maximum interval of existence, 7, 1 3 , 40 Method of Undetermined Coefficients, 23, 7 1 Meyer, Kenneth, 1 1 3 N

nonhomogeneous term, is discontinuous, 23, 79 nonuniqueness, 1 2 norm, 1 1 6 nullclines, 1 03 0

O'Malley Jr., Robert, 25 once close, 1 5 p

particular sol dtion, 7 1 periodic, closed orbit, 1 1 1 periodic harvesting, 34 periodic solutions, 32 Perron!Poincar� theorem, 122 perturbation expansions, 86 phase line, 14 phase plane, 92 phase-amplitude representation, 67 Pliss, V., 30 polynomial differential equation, 30 potential well, 106 Predator-Prey Models, 99, 101 Principle of Superposition, 72 proper nodes, 98

s

saddle point, 96 self-sustained oscillations, 1 14 semistable equilibrium point, 1 5 sensitive dependence o n initial conditions, 39 separable equation, 9 shooting method, 83 singular perturbations, 25 sink, 14 solution, 3 source, 14 stability, 1 5, 84, 1 16 stable node, 95 , 97 stable spiral, 96 steady state solution, 76 structural stability, 1 02 T

Taylor series numerical methods, 37 trajectory, 93 transient solution, 76 turning points, 26 u

undamped pendulum, 1 08 uniqueness, 5 v

variation of parameters, 2 1 , 7 1 w

Wronskian, 58, 62 R

reduction of order, 56, 64 relaxation oscillation, 1 15

Ordinary Differential Equations: A Brief Eclectic Tour (Classroom Resource Materials)

Ordinary Differential Equations: A Brief Eclectic Tour

Differential Equation Ordinary Differential Equations

Differential Equation - Ordinary Differential Equations