Springer sanders verhulst averaging methods in nonlinear dynamical systems (springer 1985)(l)(128s)

Yoshizawa: Stability Theory and the Existence Periodic Solutions and Almost Lectures Pattern Theory, Voll.. Grenander: Pattern Analysis: Lectures Pattern Theory, Vol.. 5 Averaging over S

Fluid Dynamics

Freiberger/Grenander:

Short Course

Computational

Probability

and Statistics

Operators

Hilbert Space

Wolovich:

Linear Multivariable

Systems

Berkovitz:

Optimal Control Theory

Bluman/Cole:

Similarity

Methods

for Differential

Equations

Yoshizawa:

Stability

Theory and the Existence

Periodic

Solutions

and Almost

Lectures

Pattern Theory, Voll

Driver:

Ordinary

and Delay Differential

Equations

Courant/Friedrichs:

Supersonic

Flow and

Shock Waves

Theory

Grenander:

Pattern Analysis:

Lectures

Pattern Theory, Vol

Rarity and Exponentiality

de Veubeke:

Reid:

Sturmian

Theory for Ordinary

Equations

Grenander:

Regular Structures:

Lectures

Pattern Theory, Vol

Carr:

Applications

Centre Manifold

Theory

Bengtsson/Ghil/Kallén:

Dynamic Meterology:

Data Assimilation

Spaces

Lichtenberg/Lieberman:

Regular and Stochastic

J.A Sanders Department

of Mathematics

and Computer

Science

Free University

3508

TA Utrecht

The Netherlands

1007

MA Amsterdam

The Netherlands

tions:

34A30, 34B05

34C15, 34C29

34C35

AMS Subject Classifica

Library

of

Congress Cataloging-in-Publication

55 85-22162

York Inc

©1985

by Springer-Verlag

New

All rights reserved

No part

of this book may

without written permission

from Springer-Verla

New York

10010,

U.S.A

175 Fifth

Avenue, New York,

Printed and bound

by

Halliday Lithograph,

West Hanover,

reproduced

any form

would have covered born

however,

several

books appeared

which explaine

good unders

eof the

eometry behind

existence

periodic solutions

as

more classica!

Par npvotic

analysis

has

also

been omitted

(see e.g

(Hal69a)

erations and

number

of people have kindly suggested

references,

ter toms ane

corrections

In particular

we are indebted

to

R

Cushman,

J s ister

To

reh, and S.A

van Gus

var

ar

Aen ists

we eee

lo

in versions

of the book for courses

at the

‘University

of

| U tec

ene

“Free University,

acknowledge

the generous

of the Department

of Mathematics

in Amsterdam,

of Utrecht,

and the Center

for Mathematics

and Comp

5 Averaging

over Spa

Systems

with slowly varyin

Einstein pendulum

Higher orde

Generalizatl

an example

tion

in

the regular case

Analysis passage through resonance

and examples

Normal

form polynomials dapted

of

the inner and outer

expansion,

tical pendulum

Passage through Resonance

ase;

the

Gelds with slowly

varying frequency

for motion around

an

‘oblate planet’

A dissipative

mass

References

Informal Introduction

to

Concepts

from (Global) Analysis

References

Index

220 22)

_2-

For the

Xn -matrix

A, with elements

a;

We have

here all

such generalizations

vector functions

we shall nearly

For instance

above

we pul

of the initial value problem;

f(t,x5®)

ising

in

our study

of

nà

ni cau

with respe

1.2.1

Definition

Consider

the vector function

7]

DX(0,€,)

we have

ED,

ba constant

iqueness

We

are now

theorem

for initial

Consider

the initial value

Proof

Ồy

ôy

=ô›

=0,

m which case

@()0

for /,SÍ

the right hand side

for

we have

the result

of

the lemma

to

an

we wis

— f(t) where the equation

for

has the trivial solu on)

-6- Proo

there existS

ntal matrix equ

=

we have the estimate yoryicce

On

B and

g

we

A only

r(ô)>0

such that

Poincaré-Lyapuroy

the sequel

Note that existene®

on some interval

the integral equato

x(t)

=

$()xạ

+ [8

~s+',IBG)xG)2

g(s,x(s))\ds- tat

Using the

estimates

for

©,B and

g

we have for

Using Gronwal?s

IIS Cllx

the equation

for

=x ,(¢)—x2(¢)

xi(¿)—x›zŒ)

We write the equation

lim x2(#)=0

Ix,Œ)

— x:Œ)I<Šklxiữa)

— x;ứŒ¿)Ì›

with

0k<Ì;

Ww€

shall use

-10-

value rapidly

as

long

as

” = Euler used

to Euler's example

foundation

of using

enormously

but

curiously enough only

few

authors

concerned themselves

treated

by

Eckhaus (Eck79a);

see also Fraenkel (Fra69a)

a&

=

dt

As usual,

t,fo

€[0,00);

%»%o ER"

parameter

If the

XR

the initial value problem

has

unique solution

x ,{t) for small

arising

in this

approximation

Process

can

be illustrated

by

the

following examples

Con-

sider the

first

order equation

In

defining

the concept

does not

always depend

on the param-

lems about

the demain

Such prob

order functions:

-12-

estimates

we shall often omit

‘for

<0’

The real variable

used

in

the initial value

problem

2.2-1 will

be called

time

Extensive

use shall also

be made

of

time-like variables

of

the form

r=

&et

with 5(€)=

O(1)

We

are now able

to

estimate

the order

of

magni-

tude

of functions

dg(t,©) defined

in

an interval

constant

such that

\loll

= O(ô(©))

for

<0,

S(e)

an order function

on (0,€o]

and II.||

norm for

= 9(6(e))

in

J

if lim 5(6)

that

we allow

realize that there are different

norms involved:

first

norm

to measure ,(t), then

norm, which

we shall usually take

to

be the

supremum

over

Of course,

one can give the same

definitions

for

spatial variables

of

magnitude

of

the error

approximating

sin(t

tet)

by sin(‘)

on the interva

for

the difference

of

the two functions

the boundaries

in

the example above

on the interval [ =[0,7}

We find

in the sup-

several formulations

and

we wish

our example

uld already cause notational

unique

Another

third order

not determined

sets

order functions

may define

establish

the relation

between

the solution

problem 2.3-1

and problem

2.3-Ì and

we can solve 2.3-2

as 2.3-2

and nonlinear

eneral still

turns out

that

for

initial value

general estimate

The associated

no better result than

x,()—y(Œ)=

Of

on the time-scale

we have

x,@)—y()=

O(ð())

the time-scale

Proof

We write

yữa)=%o,

the

following estimate

of

validity

we need more sophisticated

this formal approxima-

We consider

that

2.4-2 can

be

solved

explicitly

The solution

of constants)

as follows:

by

the

initial value

of

will

be called

a per-_

turbation

problem

in the

standard form

In general, however,

equation

2.4-4 will

be unattractive

Consider

for exam-

ple the

perturbed

mathematical

pendulum equation

in this case

be that the

transforma-

tion introduces

nonuniformities

in the time-dependent

behavior,

so there

no Lipschitz

biology involving elementary

be written

called quasilinear

the equa-

- 20-

$@)=e“@~*!

The standard

form

becomes

in this case

purely imaginary,

f g

is bounded

some

serious problems

prob-

lem may

of constants

to

be replaced

probl

t,))

The variation

of consta

one often

considers

the perturbed

2.5-6

Two independent

and sin(w(t

—

nts transformation

— to)

x=

such that

®(1,)=/

Equation

2.5-3

Note that the fundamental

—

€

a Snot

+ Y)g(t,

the

simplest form

standard

form which

the

averaged equation’

=o:

dt xw)=x

ptions

(not

all

following asymptotic

“ám

2.7.1 Example

—rsin(t +)

perturbation

equations become (2.5-10)

—8ersin?(1

+W)cos(t +W)cos(t)

line, the

asymp-

totic

approximation

has been indicated

the

derivative

The approximate

solution

roximate solu-

we depict the app

tion and the solution

the necessity

restrict

the time-scale

- 30

authors

The following counter example

crude averaging)

Consider

the equation

The equation

where

friction coefficient

approximation

and solution obiained

c=0.]

The numerical

-

the time-scale

figure 2.7-4

we depict z(t) and x(t) obtained

-34

We could have plotted

the crude and

obtained

the

Soviet Union from

the famous book

by

Bogoliubov

and Mitropol-

by

Volosov (Vol63a)

manifolds,

equations

with

retarded argument,

quasi-

almost- periodic equations

etc

See

also the

(Bog?6a)

1966

Roseau ((Ros66a),

provided

by Besjes_

(Bes6%a)

and Perko (Per69a)

the

last

paper more-

over

the

relation

between averaging

and

the

multiple time-scales

method has

been established

above

concerned

with approximations

were studied

by

Banfi (Ban67a)

gives

detailed

proof and new

proof could

problems

have been obtained

study periodic solutions

Burgh (Bur74a)

an

order function

4(e) plays

part, see

The order function

an o(1)

has

been derived

an

O(e”) estimate

O(5*) estimate

finds under similar

3.2.3

Lemma Consider

are for diff

the vectorfeld

ƒựứx) with

Whenever

we estimate

such

that the

estimate holds

the Lipschitz-constant

we introduce

and

we estimate

these quantities

lemmas enable

then x(t)=y(t)+

formulate

theorem, based

ƯŒx)~/"G)r|S c3

from which

estimates

Note that

values given

Proof This

standard computation:

Consider

initial value

small additions,

periodic case)

a smooth vectorfield

define

as

in Lemma

case v(t)=u(z),

so

we have from Lemma’s

3.4.2 and 3.4.3

define {„:D—>R”

€D°

Then IW(x)—„@)II=llWœ)—

and Gronwalfs

Lemma,

we obtain

second proof

The same

type

of error

0<x,<1

We ñnd

b„(x)sinQÀ„0)]|

n=\

with À„¿>0

We have:

3.4.5 have been satisfied

for the

resulting equation

Mitropolsky

devoted

book (Mit65a)

examples

Some problems

with slowly varying

time

célestial mechanics

pendulum

with slowly varying length and

some

other per-

one, and

put 1=I(et)

introduce

another time-like variable

such that s(t) increases

monotonically

and that

(O55

equation becomes

ax ge!

form cos(wot

+)

This inspires

interesting

studies have been

general theory

present form,

(See

next chapter)

Here

shall treat another

natural extension

which

applies frequently

prac-

tice

One may wonder whether sometimes

results

longer time-

over

=0

Clearly, x(t)=x,

O(€)

the time-scale

-

We shall show that u(r), the

the

formal computation

of

Lemma

5.2.1

From equations

5.3-(1-2)

we have

¿()=d;(0)+x;(03X, yiŒ)=x¡(0)+eœ,

+27;)cos()]

P=)

“lar (273)

(272-71 Jcos(p)+.a3(271)7

the equation

=0 hyperplane,

orbits

general position,

Figure 7.5.6-1 Action simplex

for

1:2:3-resonance

Apart from the usual two integrals

other integrals

quadratic

cubic third integral

exists

Discrete symmetry

p2,42 implies

been discussed

and ideas

trying

relate

Newton's theory

small intervals

Springer Verlag (911) ptic Integrals

We

shall give

here

formal derivation

references

the literature

We consider

works

fine with

heteroclinic

connections

too)

We denote any solution

and

for stable

and unstable)

From the invanant

are not uniformly

bounded,

but one can derive exponential

estimates

Let

x=x?“@)

and define

the area spanned

the planar vectors

and y)

We

Hopf Bifurcation

tion (A7-11)

such

oblate planet

161

Bracket Poisson

134,

136

Bundle circle

158 cotangent

Critical inclination

Perturbation) Duistermaat