Survey of regular and chaotic phenomena in the forced Duffing oscillator

Survey of Regular and Chaotic Phenomena in the Forced Duffkg Oscillator YOSHISUKE UEDA Department

of Electrical

Engineering.

Kyoto

University.

Kyoto 606, Japan

(Received11 March 1991)

Abstract-The

periodically

forced Duffing

oscillator

x + IcX + x3

=

Bcosr

exhibits a wide variety of interesting phenomena which are fundamental to the behavior of nonlinear regular and fractal dynamical systems. such as regular and chaotic motions. coexisting attractors. basin boundaries. and local and global bifurcations. Analog and digital simulation experiments have provided a survey of the most significant types of behavior; these experiments are essential to any in complete understanding, but the experiments alone are not sufficient. and careful interpretation terms of the geometric theory of dynamical systems is required. The results of the author’s survey. begun over 25 yr ago. are here brought together to give a reasonably complete view of the behavior of this important and prototypical dynamical system.

1. INTHODUCTION Various fascinating and fundamental phenomena occur most rcprescntative and the simplest nonlinear systems forced nonlinear oscillator governed by d’x dl?

+ k $f- + f(x)

in nonlinear systems. may be the damped,

= e(r)

One of the periodically

(I)

where k is a damping coefficient, f( x ) is a nonlinear restoring term and e(r) is a periodic function of period T. This equation, first introduced by Duffing [I], has been studied both theoretically and experimentally by many researchers. From the phenomenological point of view, a steady state governed by Duffing’s equation (1) may be a periodic motion, the fundamental period of which is either the period T of the external force, or its integral multiple. In more general dynamical systems, a steady state may be an almost periodic motion; however in the case of equation (l), the positive damping k eliminates this possibility [2]. Therefore, for the system under consideration, the regular motions are the periodic steady states. Regular motions have been extensively studied for more than 50 years. However, due to the completely deterministic nature of the equation, the possibility of chaotic motions was overlooked for a long time. The characteristic property of chaotic motion is that its long-term behavior cannot be reproduced in repeated trials from apparently identical initial condition. This contrasts dramatically with the perfect short-term predictability which is guaranteed by the deterministic nature of equation (1). Even until the beginning of the 1970s. a prejudice existed that there can occur only two kinds of steady states in the second-order nonautonomous periodic systems, that is. periodic and almost periodic motions. A similar prejudice existed among physicists who conjectured that fluid turbulence is a complex form of almost periodic motion. This belief IVY

was sharply governed

challenged by strange

in 1971 by Ruelle attractors

and Takens,

are a more

likely

author had already observed an abundance by this time, reported in [-I]; this work Ruelle’s book

articles

in La Recherche

of Thompson

and Stewart

The following Sections is essential, but on first results

of analog

Before

entering

differential

into

equations

The equation

The Mathematical

Intelligencer

and their

of

results

discrete

d.r dr

=

for the Duffing

dynamical

of the second

(1) is a particular

[5. 61, and later

in the

A BRIEF ISTRODLCTION

theory

let us briefly in

relation

explain to

the

nonlinear

order.

X(r.

d I’

z

x. _v),

= Y(r,

periodic assumccl

periodic

in to

.\‘,

system,

?‘)

t with period guarantee the

uniqucncss of solutions for any initial condition and for all f 2 [,,. A solution of equation (2) dctcrmincs ;I motion of a rcprcscntativc plant. Let us consider the solution .Y = I([;

TO

[2. 8. 1 I-171

oscillator.

systems

case of a nonautonomous

where X(t, x. y) and Y(r. x, y) are both continuity proper-tics of X and Y arc

l,,, .+Cl,.!B,,).

?’ = ).(fi

t,,. .\‘,,, y,,)

7’. Hcrc esistcncc point

sufficient and the on

the

.\‘I’

(3 1

f = f ,, is at itlC arbitrary point /j,,(x /,. y,,) of the xy plant. The in the r.ry space. and the projection of this solution curve on the .r_~ plane reprcscnts the phaseplant trajectory of the motion star[ing from I’(, at [ = f,,. Let LIS focus our attention on the location of the rcprcscntativc point l~,,(_r,,, y,,) at the

solution

instant

(2) which

the

systems Professor

interpretation.

DYNAI\lICAL SYSTEI\IS THEORY

particular

-

of equation

motions

In fact.

of irregular behavior in second-order achieved wider recognition through

OBSERb’ATION OF THE PHENOhlENON:

concepts

[3].

[2].

simulations

DISCRETE

fundamental

that irregular

of turbulence

2-5 contain some mathematical preliminaries [2. 7-161: Section 2 reading, Sections 3-5 may be skipped. Sections 6-S present the

and digital

2. STROBOSCOPIC

and

who suggested

explanation

(3) dcscribcs

[ = I,, + nT,

when

a curve

II being 0, I, 2.

An infinite (111,. /‘I.

I’>,

point

. .}

sequcncc

(J)

where x,, = I((,, + nT: f,,. I,,, J.,,), r’,, = I + 117‘; t,,. _T,,, ,b.,,) is called a pobitivc halfscqucnce or half-orbit of I),,. This half-orbit rcprcscnts the behavior of the motion starting from I),,. that is. as time proceeds, or as /I tends to infinity, the point [I,, approaches. through the transient state, the set of points tvhich rcprescnts steady final motion. J.I~ accumulation point of the positive half-orbit (4) of /I,, is called an cl,-limit point and ;I set of such points is catlcd an tulimit set of I>,,. An m-limit point and an m-limit set of ,I’,, arc ;IIXJ defined, with rcfercnce to a negati\,c half-orbit ivith II = 0. - 1. -3. . .,. An orbit i\ ;I negative half orbit to II,, plus the positive half-orbit from II,,. An orbit togcthcr uith it> (Iand tplimit sets is called ;I complctc group. This stroboscopic observation is illustrated in Fig. I by putting I,, = 0. The constant I,, rcprescnts the phase of the stroboscopic otxer\,ation and it can Ix any chosen valut~ betL+xxn 0 and 7‘. The choice of I,, may change the locations of stroboscopic points of the orbit but clots not alter their topological structure. AIW it should bc noted that due to the pcriodicity of A’ and Y the translation of the time‘ asis b>, an arbitrary multiple of 7‘ doc5 not alter the situation.

Regular

This

stroboscopic

dynamical

theory.

on H’,

or a fllilppiilg

whcrc

p, = f’i,(l~,,)

contained and rlth

oixrvation

systems

in X

of

one-to-one

continuous

f3y

or

situation

Bv which

/‘;:I.

the mapping thus

11,,(.r(0).

less

This

the

of

discrctc

dynamical

system

[I?,

point.

is illustrated plane

plane

period

I>,,

(2) with

by p,, = f;‘(f),), of the solutions

a homcomorphism, Under

that

sufficient

mapping the

is a fixed

has

a set of paramctcrs of is,

a

smoothness

and its

inverse

have

orientation-preserving.

investigate

y(O))

an nr-periodic

of the xy

is,

the successive

I = T. the equation

product

is

inverse.

is always

to

the points

situation

terms

mapping

the properties

(5)

in the xy

If a solution

ttlT,

is called

group.

I = 0 with

is the Cartesian

2.

than

and 1, denotes

From

that (5)

to study

T. then the point in Fig.

in

a discrctc

the inverse

mapping

introduced

WC have only

point

write

a continuous

system

Each

idcntifving

the

dynamical

an t,l-periodic

the mapping

is a diffeomorphism,

Finally.

the discrete

is illustrated

mapping

with

bc stated

dcfincs

itself,

We also

that

can (2)

by p,, = /T(p,,).

known

mapping

ttzT but not of period called

is

mapping

the

has period

(2).

mapping

in the IX,V space,

_V_Vplane, j,(r))

this

it

derivatives.

applying

phcnomcnon of equation

into

plarW

of the mapping

equations,

curves

Xy

Y of equation

differential assumptions,

the

the

solution

is an image of p,, under

and

iterations

continuous

of

The

and chaotic phenomena

bchaviour

images into

itself.

point

of

of initial

the

solution

points

If a solution

of the mapping

in the (x(l),

fA. This

mT, that is. a solution of period . . p,,, arc all fixed points of the and the

in Fig.

totality

3 for

of these

can be transformed the circle

points

is

ttl = 2.

rcprcscnting

to a phase periodic

space time.

(a)

1

I

PO

0 XY

XY

(b)

This

(b)

shows

that

the

stroboscopic

orbit

is ;I Poincard

stroboscopic

Unclcr

rather

the .YJ’ plane by crossing Moreover,

it from

any

outside

r,,

must

subsequent If

T,,

fi r,,

interior

to r,

The

in

this

the

plane.

exterior

x)’

plane

Therefore

of

curve

(2)

a

to the curve

there

into

sufficiently

remote

r,,

to

the

XIOTIONS exist

map.

and

the

[H, 17-l’)]

simple

closed

any one of these

the domain in such from

I-,, can bc found

through

PoincarC

in f.t-J sp;icc.

intersect

can

can be constructed

21 curve

pass

example

in practice,

of equation

property

eventually a simple

all points

choose the .ry

with

au

SET AND CENTRAL

satisfied

solution

is

origin

such

that and

that

and

all

it passes

encircles

solutions

remain

in

onI>,

to the curbe.

a manner

the

interior

interior

curve>

CLIIWS

the

starting

inside

for

all

time.

denotes

mapping

;I

the domain

point

of the I)-

Conditions

that

n curve

through origin

general

such

f‘,,

of the solution

INVr\HI,\NI’

IWUNI)E:I)

3. ~I,\SI\I:\I,

map

section

such

that

plane

by A,,. set

curve

to

the interior

= j”,(r,,).

closed

closed

exterior

A,,

of this

type,

transferred

it follows into

A,, of I‘,, is mapped

If f;(r,,) then

r,, are

is denoted lies

that

interior

under

one

interior

to A,,_,.

iterations

of the

In practice

we ma\

iteration

by I-,, and the closed

in the domain

under

points.

to the domain

domain that is.

bounded

A,

by I‘,, in

A,, C A,,,_,

holds.

z

A=n i\ called A

the maximal

similar

dissipative

bounded

construction at

large

was

invariant given

displacements.

A,,

set. by

or

Lovinson class

I):

for hc

;I

clasc

showed

of that

slxtcms the

(2)

masirn;lI

which

art’

bounc!cd

Regular and chaotic phenomena

203

invariant set so constructed is closed, connected, and has the property that images of a point which is not contained in A tend to A under iterations of the mapping fl. This invariant set A contains all the dynamics of the steady states in the system (2), however, it contains points representing not only steady states but also some transient states. Let us introduce the concepts of non-wandering set and central motions introduced by G. D. Birkhoff. It was Birkhoff who first investigated the comprehensive mathematical description of steady states in nonlinear systems. Consider an arbitrary connected region u on the xy plane. It may happen that (T is intersected by none of its images . . ., (7-2, u-1, 01, 02, . . .

(7)

where u, = f;(a). in which case u is called a wandering region and its points wandering points. Obviously. wandering points represent transient states: let us denote the totality of in no wandering points of the xy plane by W'. A point in the xy plane which is contained wandering region is called a non-wandering point. The totality of non-wandering points of the xy plane constitutes a non-empty closed invariant set M’ towards which all other points of A tend asymptotically on indefinite iteration of fA or f;‘. Let us suppose now that M’ is not identical with A, and let us take the set M’ as fundamental instead of A. A connected region which contains points of M’ will be called wandering with respect to M’ if the set u n M’ of points common to u and M’ is intersected by none of its images under powers of fA or fi’. The points of M’ which are contained in such a region are called wandering with respect to IV’, and their totality is denoted by W'. The set M’ = M’ - W' consists of the points which arc non-wandering with respect to M '.In the case where M' = M', M' is called non-wandering with respect to itself. In principle this construction can be rcpcatcd; Birkhoff composed a sequence

M', M', M', ...

(8)

where each set is non-empty and a proper subset of all those preceding it. He proved that this sequence terminates with M' = M'+'= ... which therefore must be non-wandering with respect to itself. The points of M’ are called central points, and motions started from central points are called central motions. In the 196Os, C. Pugh proved a remarkable theorem, the so-called closing lemma, which has a corollary that in a generic system the periodic orbits are dense in the non-wandering set M' [20]. Hence every point in M’ is non-wandering with respect to M’, and M' is equal to the set of central motions. Although numerical studies by their nature cannot confirm this mathematical theorem, we may say that extensive numerical experience reveals no contradiction to Pugh’s results. Here let us decompose a set of central points. A point which is both an N- and *limit point is called pseudo-recurrent. The characteristic property of such a point is that it returns infinitely often into an arbitrary small neighborhood of itself under indefinite iteration of fl as well as f:‘. Birkhoff proved that the set of central points consists of the pseudo-recurrent points together with their derived set, a central point is either a pseudo-recurrent point or an accumulation point of pseudo-recurrent points. The complete group of a pseudo-recurrent point is called a quasi-minimal set. If a non-empty invariant closed set in a quasi-minimal set is identical with the quasi-minimal set itself, the quasi-minimal set is called a minimal set, and points of a minimal set are called recurrent points. Obviously, an orbit of a recurrent point, or the minimal set itself represents a single final motion or the smallest unit of a theoretical or an ideal steady state.

4. FIXED Let

us consider

the

number

the

simplest

the classification

of these type

successive

stability

AA-D PERIODIC

points

of

imapes

of the associated

This

f,..

situation

varied For

is

values

periodic

the

in

relation

<,,a

T; I,,.

:,,.

J’,,

.\‘,, +

with

t,, = 0.

out

an ellipse

higher

than

c = (3?~/3$,,),,

terms the first

(2).

A.

The

point

type

and

These

are

behavior

of

determines

the

-

point

complex.

their

product

‘I,,.

f),,,

_r,, + r/l,)

As

shown

tl -

the

figure.

p,, in the same po\ver series

Equation

(IO)

cli, is as cl,,.

in <,, and tlii.

(*),,

dcnotcj

side

of equation

dcscribcs

mapping

Lvhen sense

the

the mapping

is charactcrizcd

ly

= 0.

the

(II)

1’

from

the quadratic

Howcvcr. is always

into

(Y)

i

in

around

and this

I)

p

‘I,,)

+

gi\,cn in the right-hand

in zC, and

of the fiscd

conjugate

equations,

to the set

and ti = (a~v/s~~,,),, whcrc

not explicitly

roots /j, and 1): arc dctermincd

diffcrcntial

equation

a periodic

T: I,,. .r,, f

1'

arc

according invariant

results

-I(n)

(I

else

[17. 211

Let p,,(.r,,. y,,) be a fixed point and ql,(.r,, + <(,. F(, + r~,) be an image of q,, under

traces

h = (a.r/Jt~,,),,.

11, and /I: of the ccluation,

the

of

or

4,

at
roots

or

solutions

Fig.

image

Q, in the neighborhood

real

points

bounded

of a fixed

solution.

PROPERTIES

point of p(,. Let q,(.r,, + $.

r/,, -

Sirlcc

maximal

of <,, and r],,. ;“, and ‘1, can be espanded

arc of dcgrcc

(IO)

its

and periodic

in the

among

the following

/I,,.

(I = (3x/3:,),,.

val11c of (*)

sets

illustrated

to encircle

small

tvith

then

of fixed

neighborhood

J’,, + q,) be a neighborin? the mapping

AND RELATED

contained

minimal

in the

POINTS

in

this

positive.

equation.

case,

bccausc

from

they the

are either

general

the following

both

theory

rclntion

of

holds.

(I?) \+,licrc (*),, The both

fixed lo,/

simple sink

dcnotcs

the value

point

/I,, is called

and

fix4

jf~?! is unity.

points

stable

or completely

saddle

or directly

saddle

or invcrscly

the point

a fiscd point

point

under

asvmptoticallv

if both

mcaris

that

if

solution

uridcr

consideration.

lp,/ and //I~] are diffcrcnt ttic

fixed

point

from

is muttiplc.

unity.

Lcvinson

If orit’ or cla5sificd

stable

IfI,l

if if

also

applies thcrc J(a)

application

solutions

I. /[I?/ < I 1, //‘?j > I 0 < 02 < 1 < (I, ,& < - 1 < p, < 0.

if

unstable

in Fig.

repeated

//‘,I <

unstable

is a sink.

as shown

the corresponding

this

unstabtc

?‘hc same classification If

at the periodic

simple

as follows:

or complctcly

source

of (*)

tend

in the scnsc

>

to periodic

points.

is a neighborhood

and (b);

all

points

of the mapping to the

,f‘l. This

periodic

of Lyapunov.

of the in this

solution, If

:I

fiscd

point

which

neighborhood

fiscd

tend

imptics

that as

SO

this

point

that

contracts to the

to

fixed

t tends to infinity. periodic

i\ a source,

solution

a neiphborhood

i\

Regular

(d)

(cl

(b) Fig. 4. Behavior

205

and chaotic phenomena

of images in the neighborhood of a periodic solution according to stability type of the fixed point. (a) Schematic r.ry diagram: (b) sink: (c) source: (d) saddle.

of the fised point expands from the point as shown in Fig. 4(c) and all points in this neighborhood move away from the fixed point under the mapping fi. If a fixed point is a saddle, we have the situation of a neighborhood which expands and contracts locally under the mapping fA asshown in Fig. 4(d). In this case. there abut at the saddle four invariant curves or branches: two n-branches, whose points converge toward the saclcilc on iteration of /;‘7 and two tubranches, whose points converge toward the saddle on iteration of fI. For the saddle, the difference between the directly and inversely unstable cases is illustrated in Fig. 5. Here succcssivc locations of a periodic solution of equation (2) through one period are shown corresponding to a directly unstable fixed unstable point I. Uy choosing the strobing angle at 12 different to T, we follow the rotation, expansion and contraction of the the phase plane at I = T Note that for purpose of illustration, position

at I = 0, so that the final image of D (or I)

point D and an inversely values progressing from 0 local N- and tubranches. is slightly shifted from its

may bc distinguished

from the initial

image. In each case, the circular dot identifies an expanding n-branch and the square dot a contracting ctrbrnnch. In the case of D, there is one full rotation during the period O-T, while in the case of I there is a half-rotation over the period. This is only the simplest case; in general, D might make an integer number of rotations during, one period, and I might Only the local linear portion of the a- and make a half-integer number of rotations. tubranches Levinson

are shown; larger portions

would exhibit

curvature due to nonlinearity.

and Massera have discussed the number of fixed points and periodic points of

equation (2) in the .r)l plane. Let N(n)

be the total number of n-periodic

points and C(n)

the total number of completely stable and completely unstable n-periodic points. Similarly, let D(n) and f(n) be the number of directly unstable and inversely unstable n-periodic points, respectively. When equation (2) has a maximal bounded invariant set and all periodic points are simple, we have the following. For rr = 1, C(1) + 1(l) N(1)

= 20(l)

= D(1)

+ 1

+ 1,

(13)

706

Y~sH~SCKE

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

UEDA

J

t=T

xv:

(a) diagram illustrating

Fig. 5. SchematIc

For

For

rz = 2. 4, 6.

(b) the difference between the two types saddle: (b) inversely unstable saddle.

of

saddlt~

(a)

Dlrectl>

unrtable

. . .. C(n)

+

I()Z)

=

N(/I)

= 2[D(n)

D(n) +

+ 2/(rz/2),

/(n/2)]

(14)

1

/I = 3. 5. 7. . . . .

(15) These

relations

points

or

conform

an indepcndcnt

points

to these

of course fixed

give

periodic

relations,

the converse

and periodic

After

and periodic

curve

in

cannot

occur

in ;I system

the

and instead

systems

refer

other systems

sadcllc structure

the

of

into

W’c have already we consider

or

the next PoincarLi

uniformly

the render

of points

are additional might

search

;I numerical type

for

observed

points

fixed do not

to be located;

be satisfied

for

a partial

but

set of

POINTS

equilibrium

order,

describe explained

the totality

f‘y

Here this

noted

set

above.

so we shall

AND RELATED

is an invariant

this

phenomena

not discuss

points.

may

it further

they

HORYZWI-.

in

let us introduce

situation one

may

only

order,

invariant

introducing

terminology

no t~vo

approach

non:iutonomous

occurs:

another.

some

[7, X]

of the second

and

different intersect

l’ROPEHTII
equations

each other,

a somewhat

mapping

damping.

differential

at the

type of minimal As

positive

by autonomous the

simplest section.

to [l7].

ASY~lIYTOTIC

the dynamics. which

points,

can intersect

of the second points

there

and

the relations

X_V plane

asymptotically

by PoincarC,

with

described in

of whcthcr

number

undoubtedly

stroboscopic

5. DOUIILY In

the

points.

the fixed

trajectories

then

If

need not be true:

closed here.

vcrificntion

is complete.

each

periodic branches

of

n complicntrd

and concepts

defined

complesity.

the (I-

and trpbranchcs

of CY- and c+brnnchcs

of fixed

of the saddles or periodic

of the mapping

points

of all orders

fi’.

If

in the

Regular

and chaotic

phenomena

20-l

xy plane, it is readily shown that no (Y- (or U-) branch can intersect another (Y- (or W) branch. However, an o-branch may intersect an wbranch, and the points of intersection in such a case are called

doubly

asymptotic

points.

A doubly asymptotic point is called homoclinic if the (Y- and cubranches

on which it lies issue from the same point or from two points belonging to the same periodic group. A homochnic point of the former type is called simple. A doubly asymptotic point is called heteroclinic if the N- and cubranches on which it lies issue from two periodic points, each of them belonging to different periodic groups. Also a doubly asymptotic point is said to be of general type if the two branches which intersect at the doubly asymptotic point cross transversely, that is, are not coincident or merely tangent at the point; in the contrary case the point is of special type. When a mapping has no doubly asymptotic points of special type and no fixed or periodic points of multiple type, the mapping is said to belong to the general analytic case. Birkhoff proved the following theorems. In the general analytic case. an arbitrary small neighborhood of a homoclinic point contains infinitely many periodic points. In the general analytic case, an arbitrary small neighborhood of a homoclinic point contains a homoclinic point of simple type. In order to illustrate these situations, Fig. 6 shows an example of a homoclinic point of simple type, together with sketches of portions of the a-branch (thick line) and the cubranch (fine line). The existence of a homoclinic point N implies the existence of additional homoclinic points H,, Hz, H3, ..., and indeed an infinite sequence of homoclinic points approaching D along its to-branch, each point being the image of its H _,,H_? and so on are homoclinic prcdccessor under fA. Likewise, successive pre-images points approaching D along its n-branch. Near D, the stretching and contraction of a neighborhood of H is governed by the local linear approximation of fI; the images of such a neighborhood, which arc schematically shown by shaded regions, become extremely long and thin under iteration of fi near D. An example of a periodic point with images pl, pr, I>~, p4, ps, /jh is indicated; additional periodic points of longer period lie nearer to the homoclinic points. The original terms N- and w-branch have been retained for reasons of nostalgia; in current parlance, the m-branch is called the unstable manifold, while the w-branch is called the stable manifold. Another alternative and perhaps more suggestive terminology was proposed by Zeeman, who calls the a-branch the outset and the w-branch the inset [22]. Up to this point, WC have considered a mathematical description of behavior of an ideal dynamical system which is perfectly deterministic, not subject to any noise or disturbance, and described with total precision. In what follows, we will investigate the behavior of this ideal system through real-world simulations using analog and digital computing devices. This brings unavoidable additional influences such as small amounts of noise, imperfect precision, and numerical errors due to

approximate solution algorithms and roundoff. Although it may not be easy to incorporate these effects into rigorous mathematical analysis, these features of real-world simulations have a healthy influence in guiding our attention to the aspects of the invariant set which correspond

to robust

6. COLLECTION

As a specific example

and stable

behavior.

OF STErlDY STATES IN THE DUFFING OSCILLXTOK [23, 211 of equation

(2), let us consider

the Duffing

equation

ecIu;~tion rcprcxnts

in a variety of applicationc, Lvith constant restoring force .Y’ rcprcscnting the simplest form of hardening symmetric spring in a mcchanicnl system. or of magnetic saturation in an clcctrical circuit with a wturablc core inductor [25-Y/, In the derivation of such ;lri quation. sonic approximations and simplifying assumptions arc’ introduced, and small This

positive

danipin$

cocfficicnt

forced oscillations k, and nonlinear

fluctuations and noise arc ncglcctccl. One might alw consider nn apparcntl!,

mow general svstcm Lvith angular forcing frequency w diffcrcnt from one and ccwfficicnt of x3 cliffcrcnt from one. Howevt2r. such ii systcrn can al~.~~ys bc brought to the form ( 16) by nppropriatcly resealing x and I. Although there arc nclvantngcs to considcrin 6 forcing. frqucncy (f) as ;i par;imc‘tt‘r. LVC uw k and /1 as the coordinates of parameter space, and do not explicitly consider trarisforma tion to other equivalent p;Ir;lmt’tt‘r spwx coorclinatc’s such as C~J and /I. The symmetry of equation (17). associated with its invariance under the substituti~~n --.\‘--+.\‘. -\‘--+,V. 1 + i;--+ l. implies that :I periodic trajectory is either symmetric to ithclf with respect to the origin of the _VY plane or it cwsists uith another per-iodic trajcctor\, symmetric to it lvith rcspcct to the origin. Note however that the stroboscopic point\ ot symmetrically relatd trajt’ctorit’s will not nppcar to bc symmetric unlcs\ c)nc of the pair is strobtxl with phase shift z. Also the positive damping k results in the non-csistcncc of sources :ind of invariant closed curves rcprcscnting almost periodic motions; 2nd the ;frC;l

of the mwirnal

bounded invariant

necessarily zcrc). In the dnmpcd. forced 0scill;itory state‘s

are

conditiona.

observccl Figure

dependin

g on

st‘t A of the mapping 1’. deflnccl 11x ccluatic)n (Iii system the’ svstcni

7 show3 the regions

~ (yivcn

bv, equation

p;ir;iniCtcrs

t\‘pe\ of stt'ad> ;I\ well ;I\ on the inl[i;il

( 16). \,ariou\

i. = (k,

on the x-0 plane in Lrhich

/I)

I\

dift‘crent

steady

motions

Regular

are observed. numerals m/n

The regions

I. II,

II’,

III

are obtained

by both

and IV characterize

(m = 1. 3, 4, 5, 6. 7, 11 and

ultrasubharmonic

and chaotic phenomena

motions

period force.

m/n

simulations.

with

period

the regions

The reman

27~. The fractions

in which

subharmonic

or

ultrasubharmonic motion of order m/n principal frequency is m/n times the

occur. An and whose

2nz. In addition

and digital

motions

n = 2. 3) indicate

of order

is a periodic motion of frequency of the external

analog

periodic

209

to these regular

steady

states,

chaotic

motions

take place in the shaded regions. In the area hatched by solid lines, chaotic motion occurs uniquely. independent of initial condition; while in the area hatched by broken lines, two different steady states coexist. one being chaotic and the other a regular motion. Which one

occurs

depends

on the

initial

condition. Ultrasubharmonic motions of higher orders in the system. but they exist only in narrow regions and

(n = 4. 5. . . ..) can occur naturally are

therefore

omitted

in Fig.

7. Also

some

subtle

details

of

region

boundaries

are

not

represented. In order to clarify the meaning of this kB chart. we choose a set of typical parameters A = (k, B) from each region. Their locations are indicated by letters from a to 11 in Fig. 7. Figure S shows the steady states trajectories observed at each of these 21 parameter values. In the figure we can see multiple steady states for certain parameter values. On the trajectories.

a x marks

the location

of a stroboscopic

point

t = 2ia (i being

at the instant

integer). Therefore these x marks show the sinks of the mapping f; (R = 1, 2. 3) in all cases whcrc the attractor is a regular periodic motion, that is, all cases except (k). (I,) and (01). The

three

drawn steady

nftcr the transient stntcs states is that, however

trajectory structure To xc

cases (X-). (I,)

and

(0,)

show

chaotic

have died away. small simulation

motions

in which

The rcmnrknblc error may bc,

the

trajectories

cannot be rcproduccd in repeated expcrimcnts; but ncvcrthclcss always eventually dcvclops. This is seen most clearly by stroboscopic the situation

more

clearly,

only

one trnjcctory

steady state stroboscopic orbit is shown in obtained indicate the prcscncc of chaotic dcfinitc structure and also an aspect of waveforms obtained by analog simulation, outc~mcs

of a random

was computed

are

feature of the chaotic the precise long-term the same sampling.

for a long time

and its

Fig. 9 for these three casts. The orbits thus attractors, that is, steady state motions with randomness. Figure IO shows corresponding which may bc impossible to distinguish from

process.

As shown above multiple attractors arc common in the system (17) and indeed there may bc regular and chaotic attractors coexisting, for example, for the cases (I) and (0). Here WC would like to associate each attractor with the enscmblc of all starting points that settle to

it;

this

point

set

is called

its

basin

of

attraction.

These

basins

arc

separated

by

r+branchcs of sonic saddle points in the xy plane, and togcthcr the basins of all attractors and their separators (basin boundaries) constitute the entire xy plane. Figure 1I shows attractor-basin phase portraits at successive times differing by phase n/6. for the parameter values (0). In this case the masimal bounded invariant set consists of the follo\ving: a sink (circle). a directly unstable saddle (filled circle) together with its a-branches, and a chaotic attractor. The non-wandering set hl’ is the sink, saddle and chaotic attractor. In the sequence shown, one folding and in the interval from I = 7r to f = 2;r a second,

and stretching action is completed, symmetrically related folding and As will be seen later, the chaotic attractor contains exactly three directly unstable type and two of invcrscly unstable type. the statistical time series analysis for a chaotic attractor. To this motion (X(I)} as a periodic process {X,(r)} with a sufficiently is a multiple of 2~. Note that in this section only, ST use T

stretching is accomplished. saddle fiscd points. one of Hcrc, Ict us proceed to end. WC regard the chaotic long period 7‘. where T esclusively to stand for this very long observation

interval,

and not

in the previous

meaning

of the period Now

of the system.

supp~sc

realization

x,(f)

x7(f)

espnnclcd

into

of

to

which be

{X,(f)}

Fourier

is hcrc

a periodic in

scrics

the

taken

equal

function

interval

to 2q.

with

(--T/2.

period T/2].

T, which

Then

the

coincides

rcalizntion

with x,(l)

a is

as &

x,(l)

= 7

+

x

(fl,,, cos 1ll01,,/

+

h,,,sin

2x 01,) = --y7

~ftu,,l),

,t,7 I

(1s)

whcrc

ItI

Fourier x7.(r) nl,,(f)

cocfficicnts is II sample and

the

(I,,, and function

average

=o,

h,,, can

of the

power

not

process

spectrum

fC~IIows. nr,y(f)

=

(X(r)>

=

f$

(X,(f))

1,2.. be

1

. . distinguished

(A’,(C)}. ‘Ij,V(~~~) of

From the

from these

process

random

varinblcs

coefficients,

the

{X(r)}

be estimated

can

bccaux

mcnn

~aluc as

211

Regular and chaotic phenomena

2.0, 1.2. 04. ZI-0.4.

-1.2.

.2.0+ -2.0

(a,)

k =

(a,) k = 0.08. B = 0.20

0.08. B = 0.20

-1.2

-0.4 x 0.4

.

3

2.0

1.2

(a,) k = 0.08. B = 0.20 20

2.0r---T-12

12

0.4

04

z1

:::L

-2.0 -1.2 -0.4 0.4 X

1.2

2.0

2,

-0.4

-0.4

-1.2

-1.2 I:

Iz'TIz -2.0 -2.0 -1.2 -0.4

X

0.4

I.2

:

-2.0 -20

-0.4

. 0.4

I.2

.

(b,) k = 0.2:. B - 0.30

(a,) k * 0.08. B = 0.20

(a,) k = 0.08. B = 0.20

-I 2

25

.,.oL---L-J -2.0 -1.2 -0.4 0.4 X

I.2

2.0

I

(c,) k = 0.015, B = 0.45

-1.5 -0.5x 0.5

I.5

(c,) k = 0.015, B = 0.45

(b ) k = 0.20. B = 0.30 2 2 5,

-25

I

2.5 -2.

2.5 (c,)

k =

0.0:s. B = 0.45

25r----i

(d,) k = 0.04, B = 0.90

(de) k = 0.04, B = 0.90

(e,)

-151

\.,_A-

i !I, i --._._,I

-251-?-?-L__-__-I 5 -05 -25 (f)

k = O.ZO,'B

05

15 =

1.50

k = 0.05.

El =

1.40

I -i 25 (9,)

k = 0.0;

B

= 3.00

SOL-40

?A

-08

(gp)

k

= 0.06.

X

08 3

24

= 3.00

810

Rsgularandchaotic

phenomena

213

a o-

-I 6-

08 2.4 X (i,) k = 0.20, 8 = 4.00

-4 0

-2 4

-08

40 (i,) k = 0.20. B = 4.00

(j,) k = 0.20, EI = 5.50

60

60

16 ZI -I 6 -6 0

-60 I

-40

-2 4

-08

08

lick-----

-I0

24

X

(j,)

IO

(k) k = 0.05:B

k = 0.20, B = 5.50

'ool----T

I 30

!

-50

= 7.50

(1,)

-30

-10

k = 0.2:.

IO

30

B = 8.50

-1

60

2.0 x -2 0 -6 0 -lOOi -50

-30

! -10

. IO

30

I

50

-5 0

(12) k = 0.2;. B = 8.50

(m,) k = 0.0:

'""r--T--60

EI= 9.50

-3 0

-10

(m,) k = 0.0:

60

60

-6 0

-6 0

IO

30

50

B = 9.50

2.0 A -2 0 -6 0 .I00

-10

-50

-3 0

-1.0

IO

30

X

(m,) k = 0.05. B =

9.50

5

o+-TT+-?-T---

-50 (n)

-30

-10

k = O.ZO.“B

IO

30

= 10.0

Fig.S. continued on p. 21-l

-100

-50 (0,)

-30

-10

k = 0.10:

IO B

30 q

12.0

50

i

-lOOl -50

-30

(oJ

-10

IO

k = 0.1;.

30

B =

I 1

-10

50

04

-50

12.0

-30

(p)

-10

I

10

k = O.ZO.xLl

30

=

4 50

13.7

(4,)

20

k = 0.07.

6 =

16.1

k = 0.20.

B =

16.5

20;

-i

x

x

+--

-20

-201

-60 1

-601

j i (q2)

k = 0.07,

B =

16.1

q3)

k

= 0.07,

8

=

16.1

(r,)

-~_

100T----_ -I60

60

-50

-30 (r2)

60N

k

-10 =

IO

0.2:.

30

E =

-10

50 (5)

16.5

k

=

O.L'O"B

10

30 =

50

IO %7--30 (t,)

18.5

-10

1

k = 0.1;.

IO 8 =

30

’ 0

18.8

I

20 x -20

t

-60.

I -1004 -50

.30 (t,)

k

-IO =

O.,:.

! IO 8

.

=

30-70 18.8

1

_.3.0 _ -10 -. -.-_IO , 30_._. 50.1

.lOly;. (u,)

k = 0.2%.

8 = 23.5

_~-30_ _--_--. -10

.,o(J, -50 (u,)

k

=

0.Z;.

10 -_ x7 9 = 23.5

-’50

Regular

-4-

*d

/

and chaotic

phenomena

-4:

215

-4-

$ I

-6

-6’

-6-

.! r!

,&Y

I -8-

-8: (k)

k = 0.05.

Fig. 9. Chaotic

attractors

B = 7.50

-8(1,)

k = 0.25,

(0,)

k = 0.10,

for three representative sets of parameter values. (Reproduced Society for Industrial and Applied Mathematics [2-t].)

(k)

k = 0.05.

(1 ) k - 0.25. 1

Fig. 10. Waveforms

B = 8.50

with

B = 12.0 the courtesy

of the

B = 7.50

B = 8.50

corresponding to the chaotic attractors of Fig. 9 obtnincd hy analog with the courtesy of the Society for Industrial and Applied Mathematics

simulation. [Z-t].)

(Reproduced

(21)

'16

4

1

I-

IO

SC

5

Ok---

h

-5

-IC!

i

I

1 -6

1

-10

, / -15

,I_

h

-siI

_;j,

/ -4

-2

0

2

4

6

1

-6

I

\\ -4

-2

3

?

4

4

X

-6

-4

-2

3 X

2

4

6

-6

-4

-2

: X

2

4

s

Regular

and chaotic phenomena

1.0

A

/

i

0.6

k = 3 -a

04

S

1 1

0.2 -

Fis. 1’.

Average power spectrum of the chaotic motion corresponding (Reproduced with the courtesy of the Institute of Electrical

The ensemble

average

can be calculated

by regarding

to the chaotic attractor of Fig. Engineers of Japan [?A].)

successive

waveforms

9(0,).

in the intervals

((II - l/2)7.. (II + l/2)7.]. (n =O. I, 2. . . ., N,) as sample functions of {X,(r)}. Let us give an example thus estimated for the representative case of the system parameters (0). that is. i\. = (k. D) = (0.1, 12.0). The mean value of {X(f)} was computed by Fast Fourier Transforms of numerical solutions over observation intervals T = 27 x 2”’ with 100 SampIeS. and WiIS found to be rn,,(/)

The mean value is found ii periodic non-stationary by using equation components

(21).

= 1.72~0s f + 0.22sin

+ 1.21cos3f

-

0.26sin3f

+ 0.25cos5r

-

0.06sin5r

+ 0.07cos7f

- 0.02sin7f

+ 0.02cos9f

- O.OlsinYf.

value

line spectra

as given

by equation

at w = 1, 3, S. . . . . indicate (22).

and numerical

line spectra represent the power concentrated at those spectra. there are continuous power spectra representing average power of this process {X(r)} is given by lim I__

(22)

to be a periodic function. This indicates that the process {X(r)} is process. Figure 12 shows the average power spectrum estimated In the figure.

of the mean

I

I.’ JT /_;-,

(X?,-(r))dr

k f

11,2 (X’,(r)) !?

frequencies. the chaotic

dr = 3.0s.

values

the periodic attached

Besides the component.

to lint The

(23)

By adding noise in the numerical integration scheme, we have also confirmed that chaotic attractors as well as averages and spectra of the corresponding chaotic motions are insensitive to numerical error or noise in the computer experiment. Therefore, it was conjectured that the average power spectrum of the process is a characteristic of the global structure of the chaotic attractor. independent of the nature of small uncertain factors or microfluctuations in the actual systems.

YOSHISL.KE UEDA

21s

7. COLLECTION

OF ATTRACTOR-BASIN

PHASE PORTRAITS

[B]

A typical example of an attractor-basin phase portrait was shown in the preceding section as the stroboscopic PoincarC section at various phases. In these portraits. basin boundaries were given by wbranches of saddles of the mapping fi in the x,v plane. These boundaries were For

located

by tracing

the example

tubranches

given

in Fig.

from

11,

the

saddles

cubranches

by reversing

were

rather

time

simple

in the simulation.

and smooth,

hence they were located with fairly good accuracy. However. as will be seen later, for the case of fractal basin boundaries, it becomes very difficult with this method of analysis to locate the basin boundaries, because tubranches become infinitely stretched and folded by homoclinic structure. therefore we cannot avoid considerable error in locating fine details of tangled basin boundaries by numerical experiment. In these situations, the full complexity of basin boundaries must be identified by exhaustive trial of various starting conditions. In order to supplement the preceding collection of steady states. let us here surve) attractor-basin A = (k,

phase

R).

portraits 13 and

In

cases,

there

the

basins

boundaries.

both

shaded

regions

regions

represent

reprcsentcd

show

by filled

six

with

points, The as

used

integrations

integration well

as

cnlargcmcrits structure similar

the

gcomctric

homoclinic which

tcncls

toward

in Fig.

narrow the final Bcforc for

stable

fiscd

invcrsclj,

point;

uristablc

The

periodic

As

periodic

basins

attractor-basin final

motions

in

The

order

can bc easily phase

of

14.

to

This

property basin in

side.

Sonic

of

shows

of

the

that

is,

of

the

the Cantor

the

a-branch the

homoclinic

small

tuo-pcrioclic attractors

the characteristic

from

\vhiclt

structure from

1-I

attractors

only

and

is

Fig.

in

a vcr-1

of a fractal

is chaotic

is tltc

t+br;irich

chaotic exist

property

the boundary

from

crosses

small

set and

boundarv

Fig. 4 of (311. Also omitted

these

caption

scqucncc.

basin

of

point.

succcssivc

origiriatcs

figure)

xtcp

grid

in each figure

boundary

the

and all fixed

each grid

cnlargcmcnt

the saddle.

cstrcmcly that

this

(saddles).

with

a scqucnce

clearly

end

of

region:

started

basin

conscqucrttly

indcterminatc. We

portraits.

use the following or

clirrctl~

the Fig.

basins S.

from

arc marked

group

but

movcmcnt

harrtlonic

point:

ones

in

groups

sink

our

symbols

or completcl)

(U)

for

the

basin

saddle

by the S;IIIIC

of itnagcs

under

casts

in Fig.

arc

symbol. points

the S.

(c) art the k/I

and Llltrasub~larnlolli~s

boundaries

(attractors)

periodic

to the point

explain for

or

01 = 2. 3).

related

tttc corrcspondirig

c~)rrcsponding

arc the fundamental

and

periodic

here

(0)

fiscd

point

sinks

of diffcrcnt

of succcssivc xccri

arc

multiple

we must symbols:

unstable

(X. +) for II-periodic

basins

portrait

basin

figures.

a uniform for

arc given

Fig.

shown

right

Note

to the SIII~C

complcsity,

the

blank

saclcllc

inside

sho\\n

points

of the

(not

attractor-basin

point;

bclongirtg

to avoid

distinguished. thcsc

points

points

for

nature

a pair

behavior

points.

(0)

no

on

was confirmed

In

6 of (301 and

v~~lucs.

becomes

fiscd

was

of

fractal

points

these

chosen

of the _V_Vplane is

branches

from

the

the other

and periodic

saddles. order

basins

motion

presenting

of grid

attractor Fig.

parameters

and

scheme

wcrc

behavior

infinitely.

fractal

invariant

inside

is th.at transient

the fised

The

system

fixed

Rung-Kuttn-Gill

final

regions

unstable

In computing

conditions

Thcrc

saddle

of pararnctcr

steady

smaller

of [29],

dctcctcd

range

boundary

the

l(b)

directly

paramctcrs.

continues

the

csist

boundaries.

numbers

system

non-rcsortatit

narrofi’

wcrc

until

boundarks.

of

the

art’ cstrcmcly

arid

basin initial

the

the

fractal.

structure

toward

illustratccl

and

of

structure or

tcrids

which

sizes

basin

self-similar

c~illccl

continued

of

smooth

(sinks) represented by small circles; the of the resonant motions, while the blank

a fourth-order

being

sets

between

attractors

Thcrc

in their precision:

values

representative difference

of attraction

hereafter.

of smaller

of

are two

single

step

the

the

ones.

circles,

portraits

additional

14 illustrate

non-resonant

attractor-basin was

for

Figures

However. art:

not

mapping Figure chart

of orclcr

;Irc

comrtion: in

al\va)s

,/‘,, among I5 shows

of J/3.

Fig. 7. Figure

a11 and

I I)

Regular

219

and chaotic phenomena

Fig. 13. Attractor-basin phase portrait with integration step size 2rr/60 and 201 x 201 grid of initial conditions for k = 0.1. E = 0.3. (Reproduced with the courtesy of the European Conference on Circuit Theory and Design [ZS].)

_ 2.0

_.

:. -.: --.

1.0

~: -5:

x

‘-

0.0

0 -j

.o”

L’..

1.

.

I.

:.:..

I..

-1.0

-2.0

Fig. 14. Attractor-basin phase portrait with integration step size Zn/t%l and 201 x 201. 161 x 161. 161 x 161 grid of initial conditions for k = 0.05. E = 0.3. (Rcproduccd with the courtesy of the European Conference on Circuit Theory and Design [2X].)

corresponds to the point (e), and ultrasubharmonics but

it should

even,

there

odd.

only

chart

be noted appear

one

esplained Figure

and ultrasubharmonics of order 3/2. Figure 17 corresponds to of order 5/3. All of these portraits are similar to each other.

that

a pair

such

above. 1s shows

of Fig.

basins more,

(d).

for

ultrasubharmonics

of attractors

attractor

appears.

an attractor-basin

7; see also

[32].

This portrait

There

m/n,

of order

and basins,

while

results

from

the

corresponding

are two

attractors

when

in cases where

either

symmetry

to the

point

of two-periodic

rn or

II is

m and n art’

both of

the

(j)

on

groups;

system the

their

kB two

are separated by the ccrbranches of a directly unstable fixed point of fi. Furthereach basin of a two-periodic point is subdivided by tubranches of the inversely

unstable

fixed

tlvo-periodic crhbranches Figure

points.

The tails of these four subdivisions

groups respectively. of the directly unstable 19 corresponds

becoming infinitely fixed point.

to the point

(II)

of Fig.

of the two basins wind thin

as they

around

the

along

the

only

one

accumulate

7. In this example.

there

exists

attracting three-periodic group. The three images of this three-periodic point are considered as distinct attractors of the mapping fi. and the corresponding basins arc distinguished. As is seen in the figure, a more complicated configuration appears, that is. each basin of an image

of the three-periodic

unstable

points figure.

fashion. ctrbranches iClorcovcr.

three-periodic seen in the

point

is bounded

by crhbranches

of the three fiscd points tails of the basins arc mixed

(one dircctl) in confusion

ctrbranchcs of thcsc three saddles from inspection reveals that the basin boundarks

and two inversely unstable saddles). with each other and accumulate on the

both sides. It should bc added that a clo~cr of Fig. l’j have a fractal structure. but those ot’

Fig. IS do not. This means that invariant branches of the clircctly unstable IS arc hctcroclinic but not homoclinic, while those of Fig. 19 arc homoclinic. Figures stable grl)up

20 and 21 correspond

of the dircctlq

and the tails of these three basins behave in a complicated becoming infinitely thin as they accurnulatc along the

to (q)

and (I).

rcspcctivcly.

In addition

figure, it has been structure. In Fig.

of Fig.

to the complctcl)

fisctl point, thcrc txist two two-pcrioclic groups in Fig. 20 and one in f-ig. 21. Lklow the frame of Fiq. 20 thcrc esists a directly unstable L

shown in the figure. Though not indicated in the invariant branches of this saddle have a homoclinic

saddles

three-pcrioclic fixed point as

confirmed that the 20, the tr+branches

clclimiting basin boundarks of two-periodic points do not have homoclinic WC have confirrncd that thcsc basins arc caught in the homoclinic structure

structure. but of the s~tdclle

below the franic. and hcncc tails of the basins show estrcmcly complicated shilpc ;IS WC see in the figure. This cast shows a second mechanism for generating a fractal basin boundary. Taking the saddle below the frame of Fig. 20 into account, the Levinson-Masscra relation (13) concerning the number of the fiscd points holds and also equations (1-I) and (15)

can

bc verified

for

the

n-periodic

points

(n = 2, 3) for

all

attractor-basin

portraits

given above. Within cnch region in Fig. 7 identified by Roman numerals. the‘ number of fiscd points i\ constant. but the number changes across the boundaries of thcsc regions. It is believed that all fiscd points which can csist Lvithin the parameter range of Fig. 7 have bcrn identified. although WC cannot cscludc the possibility that some fised points esisting in some small at low damping may hnvc been neglcctcd. From Fig. S and the attractor-basin rt’cion c portraits of Figs 13-21 it is conjccturcd that the counts of fixed points in Tabit 1 ;IK cornpletc \\‘ithili the indicated regions. Recallin g that there arc nu sources, IVO use S( 1) to dcnotc the number of period one sinks; as before. D(I) and I( I) denote nunibcrs of directly and invcrscly unstable saddle fiscd points.

Regular

2.0 x

1.0

1.0

x

t

x

x

0.0

. 0

h

i

0.0

-1.0 I

i

_i-. . : I x

”

+ -1.0

221

and chaotic phenomena

:

x

j/.

i

~

“_~ ii

‘.

-:.

.:.

-.

:.

“,

-2.0

:

+

:

I.

:.

+

I,

:

:,,

:.,

.I

:.

-2.0

-

-1.0

0.0

1.0

3. 0

2.0

X

X

Fig. 15. Attractor-basin tion step size ?a/60 conditions for case (c) (Reproduced with the ference on Circuit

phase portrait with integraand 201 x 201 grid of initial of Fig. 7; A- = 0.015. 8 = 0.45. courtesy of the European ConTheory and Design [2X].)

Fig. 16. Attractor-basin tion step size ?n/60 conditions for case (d) (Reproduced with the ference on Circuit

2.0

-

1.0

-

-*

phase portrait with integraand 201 x 201 grid of initial of Fig. 7: k =0.0-l. E = 0.90. courtesy of the European ConTheory and Design [ZU].)

-. x

x

; ‘.

x

0

o.o-

‘,

Y

*x

’ I

-1.0 -

Fig. 17. Attractor-hasIn phnsc portrait with intcgraI?n,‘60 and 201 X 201 grid of initial lion step six conditions for C;L\C’(~9) of Fig. 7: X = O.OS, II = 1.10. (Rrproduccd with the courtesy of the European Confcrcncr on Circuit Theory and Design [2X].)

I II-II’-111 II’ II n III III-II IV Outside I u II u III u IV

2.

S(I) S(1) S(I) S(I) .S( I) S(I)

= or = = = or

S(I)

= I

I( I) = 2. 4. I. 7. I( I) = 2.

-2.0

x ‘.

-

1 -1.0

I 0.0

1.0 X

D(l) D(1) D(I) .S(I)or D(I) 1)(I)

= 1 = I = 3 /(I) = I = I

= 7. 11(i)

= 2

I

I

2.0

3.0

6.0

x

-6.0

-8.0 2.0

I

L

L

2.5

3.0

3.5

: 4.0

X

X.

Let

GI,Ol~r\l,

us

now

particularly Figure

22

shows

The

number prolongcd,

self-similar

property

cviclcncc the

phenomenon parameter stems ncnr

to the

iMost instances tanlrcncics

is not values vary

point

continuously

’ D’

the

the

as the

(0)

I~IFUHCA’I’IONS

above

in

in

two

inversely sntltllc

bc

seen.

Follo\ving homoclinic

that

light

of

the

theory.

specific

the

parameters

pnrnmetcr

arc

shoun

more’

of

this

attractor

directly of

region.

values

and

some

points

unstahlc

Birkhoff,

chaotic

of

shaded

As

plane. fisctl

of

the

contain&

in

part,

these

in

’ 1’ and

points

nncl

brnnchcs

implies

the

;I

arc

esistcncc

points.

the

manifolds

kf1

unstable

unstable can

the

three

and

directly

corresponding

intersections

intersections or

arc

suggests to

MSIN

are

case

The

is

identical

unstable

saddle

(o),

occurs

but

appearnncc

changed

inside

of the

with

the

’ I>‘, This for

the

typical

attractor

shad&

region

(0).

homoclinic of

the

case

appear.

ncnr

unstable

peculiar in

will

points or

for

intersections

strongly

n-branches

the

dcscribccl

Thcrc point

of

homoclinic

Numcricai of

attractor

unstable

/\Ni)

structures.

structure.

cebranchcs

periodic

facts

manifold

transvtzrsc many

/\‘l”I‘KAC’I‘Ol~

cxpcrimcntal

chaotic

directly

CI- and

of infinitely clusurc

the

manifold

one

of

S’I’HUCTUHE.

invariant the

invariant

attractor,

‘I’.

interpret

the global

associntctl the

,\‘I”I‘HACI‘OH

near

bvhich

tangcncics

in arc may

Fig. very

22

are

nearly

appear

clearly

transvcrsc.

tangent. as

more

It of

but

is espcctcd the

invariant

there

arc

that

many

manifolds

a ftx such art’

Regular

and chaotic

phenomena

8.0

6.0

4.0

2.0

x

0.0

x

-2.0

x

-4.0

+

+.

:

-6.0

hase portrait with Fig. 20. Attractor-basin integration step size 217r180 and 241 x 321 grid of initial conditions for case (q) of Fig. 7; k =0.07, B = 16.1. (Rcproduccd with the courtesy of the European Confcrcnce on Circuit Theory and Design [ZS].)

. 8.0

6.0

4.0

2.0

.. x

0.0

1

.

-2.0

1

x

-4.0

-6.0 Fig. 21. Attractor-basin hasc portrait with integration step size 2a /“% lb0 and 241 X 321 grid of initial conditions for case (I) of Fig. 7; k = 0.12. B = 18.8. (Rcproduccd with the courtesy of the European Confcrcnce on Circuit Theory and Design (281.)

x

-8.0 -1.0

0.0

I.0

2.0 X

3.0

4.0

‘2-I 10 -

c

I

0 x

-5

-10

Numerical

cspcrinwnts

attrx(or

arc

sniallcst

tlic

The

nearly

that

Furthcrriiorc,

evidcncc

strongly

infinitely

often

Thus

the

unstable:

of

rnotiorl

inay

Lx

but this

tcrrn

is not

is :I sin@

in

an\

clic

part

of

or

of

the

On fill

aLbay.

the

ml;111

[hat

attractor.

A i.e.

thi5

other

h:tnd.

crpprcn1l~

fhC

typical

thcrc

scnx

211ractor:

an

73~1s

the

w;lvcforn~\.

the‘

out

’ I)‘.

attractor,

the

in

so

in

rnotiom

corlclitions.

n-hranclics transitive

point

motions arc

rcproclLlcible.

diffcrcnt

g from

transients the

pcrioclic atlraction\

akv2y.

011 initial

of anj’

of

to

xlartin

nt‘tcr

inclicatcs

bq’

in

the

the

since

callcd twcn

rdncloni

thought

infinite the

small

real

WC have has

bc

art’

anwng

influenced

reason chaos

riianncr

prcscnt

may

Lvhich

molions.

random

equations

stronglv

thc’rt’

2 neighborhood

obscrvccl

pcrioclic

apparently

For

will.

that

syrcnl

cvcntuall!,

nlotiorls

closure

nlany

basin.\

iteration

dcpcnckncc

trajectory

suggests

the unclcr

Icad t’or

the

their

pcrturlx attractor

occurs

inlinitcly

the

nunlcricll

orbit

return\

i5 stahilitL

~II the

Poisson.

of

tr;insit

error the

st2nsitic.c

i4,

Of exist.

conditions

that to

all sinks

arty

or in

as

to

long-tcrni

structure.

xnsc

noise

images

sitiialion

ret‘crrtxl

single

it‘

initial

this is

idcritical

The

of of

iclcntical

property any

Thus

amount

movcnwnl

that

shmr

unstahlc.

r)l’

iit infiiiitcly

this Lviclcly aymt

or

or

noiw

s\‘stcrn

unstable

not

ot

continuc~ pcrioclic

irlcludctl

to

various trarl\it

ill

wIution\.

in

the

The

rlit‘fcrcntial

siinul:ltic)n.

type

ot’

;I ranclonily

motion

acccptcd

to

the

pIlCnoIIICII;I.

of

ri~i_chI,orlioocf\

visiting The

rn;in>’

fluctuation5

system

as

riunitwr.

dcxrihc:

thi? although

transitional

type

of it

motion

motion:

rri;i>’

not

this

[‘_>I.

tcr-m

aclquatcl~

Regular and chaotic phenomena

225

convey the very coherent structure which is an equally important aspect of the motion. Let us now turn to a description of the various routes to chaos observed as the system parameters are varied from outside into the shaded regions. The most commonly observed transition is by successive period doubling, beginning from the two symmetrically related sinks which exist everywhere inside region II. This first period doubling is illustrated in Fig. S from case (i) to case (j). The arc in Fig. 7 passing between points (i) and (j) shows the location where this first period doubling occurs. Notice that this period doubling bifurcation arc extends over and around to the right side of the shaded regions. A typical path in the kE plane which passes through this arc into the shaded regions leads to chaos via Feigenbaum

cascade,

generating

two

tually merge with each other. A different bifurcation is observed

symmetrically

related

when

the shaded

entering

chaotic

attractors

(0)

region

which from

even-

the right,

that is, from the region typified by case (p). At this edge of the shaded (u) region, a global bifurcation occurs which causes the chaotic attractor to suddenly appear. The global bifurcation is a homoclinic tangency of the (Y- and (u-branches of the directly unstable fixed point (filled circle) in the basin boundary. This situation is illustrated in Figs 23-24. The gains or parameter values k = 0.1. B = 13.355 are the values where the chaotic attractor loses stability, and also the values where the directly unstable fixed point has a homoclinic tangency. This phenomenon was originally similar phenomena have since been widely

described in [34] and called a transition chain: reported and are now frequently referred to as

boundary crises [35] or blue sky catastrophes [36]. Another example of such a global bifurcation occurs along the right boundary of the shaded region containing case (/). Here the chaotic attractor gains or loses stability as it whose CY- and c&ranches develop a touches a directly unstable three-periodic point, homoclinic tangency at the bifurcation threshold. This situation is illustrated in Fig. 25. Note that in both this and the preceding boundary has no hornoclinic structure bcforc that

the chaotic

attractor

always

unstable point cast. the directly the transition chain is established.

has a regular

basin

boundary.

In other

happen that a chaotic attractor exists in a basin with a fractal boundary, structure dcvclops in the basin boundary prior to the transition

in the basin This means

systems,

it can

that is, homoclinic chain or blue sky

catztrophc. Further clcscription of this phcnomcnon can bc found in [37], where the term chaotic sacldlc catastrophe was used: set also [3S]. The phonomcnon of chaotic sadcllc catastrophe has not been obscrvcd for equation (16) in the region of the kB plane shown in Fig. 7. Finally wc clcscribc the bifurcations passing from the shadccl region typified by case (0). Hcrc complicatccl. The chaotic attractor this period three sink experiences chaotic attractor; then Iargc chaotic attractor.

develops ;I period

the region of cast (n) to the right into the phcnomcna arc somewhat more

from the pcriocl doubling cascade

thcrc is usually a sudden explosion Similar phcnomcnn wcrc rcportcd

widely known a5 interior crises [35]; see also [ 131. WC note that in all casts whcrc chaotic attractors

three sink of cast (II). First which leads to a three-piccc

in size from a three-piocc to one in [39], and have since bccomc

are observccl,

they contain

among

the

pcrioclic points of lowest pcriocl cithcr one inversely unstable point. or one directly unstable and two inversely unstable points. Recently Stewart has conjectured that this can bc csplaincd by applying the Lcvinson-IMasscra index equations to an appropriate suhrcgion of the .r_v plnnc [JO]. Somctimcs the global structure of the phase portrait may have important conscqucnccs cvcn though all attractors arc regular periodic motions. For csarnplc. a local saddle-node or fold bifurcation causes the system to undergo a rapid transient to some other attractor: and it can hnppcn that if two or more attractors arc availnblc. the one chosen may depend

‘2 6

Y~SHISL.KE

U~D.A

Fig. 2.7. Attractor-hnsirl phase portrait with integration step bizt2 ?n/‘lXO amI 241 X i2t for I. = I). I, grid of inllial conditions

n = l?.?SS.

10.0

1

8.0

6.0

4.0

2.0

x

0.0

-2.0

-4.0

-6.0

-8.0

I

-10.0 0.0

1.0

2.0

3.0 X

4.0

5.0

Regular

Cl

and chaotic phenomena

6.0 t

_4.0[ , Fig. 25. Attrnctur-basin phase portrait with inteprntiw step SIX Za/l?O and 201 x 301 grid of initial coliditiuns for k = 0.25. H = 8.812.

1.0

I

I

,

2.0

3.0

4.0

X

very scnsitivcly on how the bifurcation is realized in the simulation or real-world system. Such inclctcrminntc bifurcations can bc obscrvcd in the Duffing equation (16). for cxnmplc Although the bifurcation cvcnt is by crossing the boundary of region I in the kf1 plant. local,

various

the outcome

attractors

is dctcrminccl

available

by global

can bc cstimatcd

structure:

the probabilities

from the structure

of settling

of invariant

on the

manifolds;

SW

[Jll. 9. CONCLUSION

By summarizing the author’s previous reports [23-24, 2S, 33-34, 391, regular and chaotic phenomena have been surveyed which occur in the system governed by the Duffing and attractor and basin bifurcations have been equation. The global attractor structure, discussed in relation to the geometric theory of differential equations. Chaotic attractors have been observed over a wide range of parameter values. Multiple coexisting attractors are also common; both regular and fractal basin boundaries arc observed. The structure of chaotic

attractors,

basin

boundaries

and

bifurcations

have

been

understood

unstable periodic motions and invariant manifolds. High resolution portraits of chaotic attractors shown in Fig. 9(k) enough to earn the attention of a wide audicncc: 0. Rucllc nnmcd

in terms

of

and (0,) were lucky the attractor of Fig.

9( o ,) Jrrpttcsc autm-[or [5, 61. and Thompson and Steward referred to the attractor of Fig. c)(k) as Ucdtr’s chotic ofrtxcfor [2]. Ucda’s chaotic attractor shown in Fig. 26 and the Japancsc attractor in Fig. 27 conclude the article. Both pictures arc dcpictcd taking the unit Icngth of the .r-axis to bc five times that of the y-axis, and 100,000 steady state points arc plotted.

Acknowltd~emmr-I owe sincere thanks to the mathematician Dr Hugh Bruce Stewart of the D~vls~on of r\ppl~ed Science. Brookhaben National Laboratory. for his generous asvstance in preparing thlb article. Not only dtd he graciously agree to edit my manuscript, which was wrltten in poor English. but he al\o provided me ulth much valuable advice in structurtng the text as well as selectin! the figures. I would like to point out that most of the items withln the text that are not specifically described tn the References [23-74. 3. 33-3-I. iU]. are largel! 0 result of his advice. I would also like to thank Dr Stewart a? well a.5 Professor Ralph Abraham of the UniverGty of Callfornla Santa Cruz. especially for their tnrisht into the significance of C. Pugh’s cloxinp lemma. which I had not fully gr;l\pcd

Regular and chaotic phenomena

Fig. 27. Jnpancx altraclor.

229

230

REFERENCES

Regular

and chaotic

phenomena

231

in chaotic attractors, and transient chaos, 35. C. Grebogi. E. Ott and J. A. Yorke, Crises. sudden changes Physica 7D. 181-200 (1983). 36. R. H. Abraham, Chaostrophes, intermittency, and noise, in Chaos. Fractals, and Dynamics. edited by P. Fischer and W. R. Smith, pp. 3-22. Marcel Dekker, New York (1985). in Dyamicol Systems Approaches to 37. H. B. Stewart, A chaotic saddle catastrophe in forced oscillators. Nonlinear Problemr in Systems and Circuits, edited by F. M. A. Salam and M. L. Levi, pp. 138-149. SIAM. Philadelphia (1988). 38. C. Grebogi, E. Ott and J. A. Yorke. Basin boundary metamorphoses: changes in accessible boundary orbits, Physica LJD. 243-262 (1987). 39. Y. Ueda. Explosion of strange attractors exhibited by Duffing’s equation. Ann. IVY Acod. Sci. 357, 422434 (1980). Jo. H. B. Stewart, Application of fixed point theory to chaotic attractors of forced oscillators. Research Report. NIFS-62. National Institute for Fusion Science. Nagoya. Japan (1990): Japun J. Ind. Appl. Math.. to appear. 41. H. B. Stewart and Y. Ueda. Catastrophes with indeterminate outcome. Proc. R. Sot. A.l32. 113-123 (1991).