PERSISTENT PROPERTIES OF CRISES IN A DUFFING OSCILLATOR

Persistent properties

of

crises

in

a Duffing oscillator

Yao Huang Kao

Department ofTelecommunication Engineering, National Chiao Tung University, Hsinchu, Taiwan 30050,Republic ofChina

Jeun Chyuan Huang and Yih Shun Gou

Department

of

Electrophysics, National Chiao Tung University, Hsinchu, Taiwan 30050,Republic

of

China (Received 2February 1987)

Crises in a two-well forced oscillator ofDuffing type are studied with an analog simulation. Its features are discussed with the aid ofreturn _{maps and phase} portraits. Two types ofboundary crisis in a strange attractor following the Feigenbaum route to chaos are found. One isassociated with a hopping between two strange attractors and is confirmed with the presence of

1/f

noise spectrum. The other isassociated with a hysteresis jump. Aborderline ofdifferentiating these two characters isalso indicated.

I.

INTRODUCTION

X+KX+aX+PX

=F

sin(cot) . (3)

Studies

of

chaotic motions in nonlinear dissipative sys-tems are

of

great interest and have received much atten-tion in recent years.

'

The chaotic motion may exhibit a crisis event with sudden qualitative change

of

a strange

attractor as a controlled parameter is varied. According

to Cxrebogi, Ott, and Yorke, the crisis in a one-dimensional quadratic map occurs at certain parameter values, for which the strange attractor collides with an unstable periodic orbit. Since then, the crisis has been confirmed experimentally in several physical systems, in-cluding Josephson junctions ' _{and some nonlinear driven}

oscillators. In particular, the analog simulation

of

the rf-driven Josephson junction performed in our previous work further indicates that the crisis induces either a hopping state with

1/f

low-frequence noise or a hysteresis jump for different parameters set. Nevertheless, the

oc-currence

of

these two effects related to the crisis isnot yet fully understood. In fact, we note that a global descrip-tion

of

persistent properties

of

the crisis in diverse physi-cal systems isstill necessary to find in order to elucidate a true mechanism

of

the crises in chaotic dynamics.

To

this end, we shall present an electronic analog study

of

the crisis effects in a symmetrical two-well potential forced oscillator, which also represents the dynamics

of

a buckled beam as well as a plasma oscillator. ' Its

dynamics

of

motion can be modeled by the following equation as

The chaotic motions in Eq. (3) have been investigated by many authors in the past few years.

"

It isknown that, at a certain situation, the motion swings chaotically between the two valleys as a hopping state. The features

of

the hopping state such as the fractal dimension

of

both the strange attractor and basin boundary, '

''

the

1/f

low fre-quency spectrum, ' and the similarity exponents' have been found, respectively. In spite

of

such considerable ef-forts sofar, properties

of

the crisis in this system have not yet been pointed out. The goal

of

this paper, therefore, is

to increase our understanding

of

the crisis event in the Duffing equation. The important roles

of

the symmetry and nonlinearity

of

the central barrier

of

the potential well in the occurrence

of

crises are also emphasized.

The selection

of

the analog simulator which is con-structed with two usual integrators and two multipliers is based on the following reasons. It has the advantage

of

fast response over the numerical calculations as the pa-rameters are varied, so that itenables us to uncover quick-ly a guideline by which a more precise study

of

the

nu-O.c—

X+

XX+

d V(X)/dX

=F

sin(tot

),

where

K

isthe damping coefficient,

F

isthe driving force,

cois the driving frequency, and V(X)is the potential.

For

atwo-well case, the potential V(X)is given by

V(X)

=

2

aX

+

,

f3X—

g(x)

0.

2-—_0.2—

with

a

&0

and /3&0. Thus, it has a central potential bar-rier around

X

=

0 and two stable equilibra at

X=+( —

a/P)'~

as shown in Fig. 1. And

Eq.

(1) be-comes an ordinary differential equation

of

Duffing type as

FIG.

1. Symmetrical two-well potential V(X)with a central barrier at

X

=0.

merical calculation can be followed. Moreover, together with a sample-hold circuit, it shows directly a return map

of

the attractor on the scope so that the detailed features

of

the crisis can be effectively traced.

II.

EXPERIMENTAL RESULTS

0.4 0.3 0.2 -FBD 0.1— -FAD FAB 0 I 0.6 I 1.0

FIG.

2. State diagram in the parameter space

F

vs co with threshold curves;

2,

A': downward jump; B,

B':

upward jump; C: period doubling; D: boundary crisis, with damping coefficient

%=0.

1. All dashed curves are obtained with an ini-tial state above curve

B.

The dotted curve

2'

is obtained with an initial state above curves

B'

and

B.

As pointed out in our previous paper, the dynamic behaviors in Eq. (3) can be properly investigated by scan-ning either the driving amplitude

F

or frequence co with coefficients K,

a,

and

P

as parameters. From variations

of

the phase portrait

X

versus

X

and/or the return map a state diagram which illustrates the thresholds

of

hysteresis jump, period doubling, and crisis in

F

versus co space is

constructed. The return map characterizes the voltage

of

variable

X

at any instant,

_X„+&,

as a function

of

the same voltage but one period earlier,

X„.

A typical case with

K

=

0.

1,

a

= —

1,

P=

1, and

0.

5 &co&

l.

4is shown in Fig. 2. In this figure there are two similar groups

of

threshold curves related to the primary and secondary resonances with co in the ranges

0.

8

—

1.

4 and

0.

5

—

0.

7, respectively.

Both behave in a global manner. '

For

convenience, we focus only on the former case. Curves

3

and A' are the downward-jump threshold, curves

8

and

8'

are the upward-jump threshold, curve C is the period-doubling threshold, and curve Dis the threshold

of

boundary crisis.

These curves intersect each other at critical points

(F~~,co~~) for curves A and

B,

(F&D,co&D) for curves A and D, and (F~D,co&D) for curves

B

and D, respectively. With reference to Fig.2, there are two types

of

both crisis and hysteresis, respectively, depending on the scanning procedures. The salient features

of

each scanning are summarized below.

A. Amplitude scanning

1. co

)

cogD

For

a typical demonstration

of

the results, we choose

co=

1.2, a value larger than cozD

(=

1.125), and set initially the state at the equilibrium point

X

=1.

0.

As the driving amplitude

F

is increased from zero (path I),a sequence

of

transitions occur on the threshold curves. The phase por-traits near these transitions are shown in Fig.

3.

First

of

all, on curve

8

with

F=0.

097, the unsymmetrical phase portrait expands abruptly as shown in Fig. 3(a). This behavior is referred as upward jump. On the contrary, while

F

isturned back from above curve

8,

the phase por-trait is restored suddenly to the initial shape on curve

3

with

F=0.

033.

This is referred to as a downward jump.

Hence, curves

3

and

8

form a hysteresis loop with the motion confined only in the right valley. Note that, in these situations, the nonlinear effect issmall.

As

F

is increased to

0.

182 on curve C, the threshold

of

period doubling, the phase portrait splits into two cycles as shown in Fig. 3(b). The state undergoes the Feigenbaum's route tochaos soon after period four as

F

is further increased. It implies that the nonlinear effect is strong enough to cause the chaotic motion. Due to a less steep slope

of

the potential well around the central barrier, the trajectory

of

the phase portrait in the left-hand side splits much more distinctly and approaches to the unsta-ble equilibrium point at

X

=

0.

In other words, the separation

of

the trajectory is essentially dependent on the slope

of

the potential well. When

F

isincreased a little bit

to

0.

198on curve D,the onset

of

crisis, the motion begins

to migrate into the left valley and exhibits ahopping state as shown in Fig. 3(c). We refer to it later as crisis-induced hopping.

As

F

is further increased, the state keeps on hopping first and then is locked to some subharmonics. Figure 3(d) shows the phase portrait

of

a typical case for —,'

subharmonic with

F=0.

268. This state exists in a small interval

of

the driving amplitude. As

F

is increased to

0.

321 on curve

8',

the hopping state disappears suddenly and becomes stable with a symmetrical phase portrait ex-tending over the two valleys as shown in Fig. 3(e). It im-plies that the central barrier plays no significant effect on the motion for large excitations. In other words, the sys-tem isagain in a situation with small nonlinearity.

Alternatively,

if

the amplitude

F

is decreased from above curve

8',

adownward jurnp occurs at

F=0.

110

on curve

A'.

The larger symmetrical phase portrait is re-stored abruptly to asmaller nonsymmetrical one as shown in Fig.

3(f).

Note that curve A' is found only when the initial state is above both curves

8

and

8

.

This unusual type

of

hysteresis with a great contraction implies a

catas-trophe associated with a symmetry breaking and ought to

receive more attention.

Now, in order to elucidate the features

of

crisis

occur-ring on curve D, the corresponding return maps as shown in Fig. 4are traced. Figure 4(a) shows the return map

of

the chaotic attractor between the cascaded period dou-blings and crisis with

F=0.

194. The map is similar to a quadratic form, however, with a back-folded tail (mark).

As

F

is further increased, the folded tail extends more and more close to the line with

X„+&

—

—

X„.

At

F =0.

198on curve D the tail meets at a point (mark) on the line

X„+&

——

X„.

_{The point is referred to as an unstable} fixed point with a slope

dX„+~/dX„

larger than 1 as shown in

Fig. 4(b). This case implies the occurrence

of

a boundary

crisis. In this moment, the motion in the right valley at

X&0

jumps into the left one at

X

&0

and, meanwhile,

develops anew part

of

the chaotic attractor with the simi-lar quadratic shape. Therefore, it is a nondestructive type

of

boundary crisis with abrupt expansion

of

the strange

attractor.

Similarly, the dynamic process

of

the state in the left valley moving to the right one behaves in the same manner asjust mentioned. Thus, a crisis-induced hopping between the two strange attractors appears. This feature

Ea)

(e)

FIG.

3. Phase portraits for

co=1.

2 and (a)

F =0.

092, the states just below (small) and above (large) curve B;(b)

F=0.

182,the period-doubled state just above curve C; (c)

F=0.

198 the hopping state just above curve D;(d)

F=0.

268, the ~ subharrnonic state;

(e)

F=0.

321,the state just above curve

B';

(f)

F=0.

110,the state just above (large) below (sma11)curve A'. Xaxis: X,0.5U/div. Y axis: X,0.5U/div.

B.

Frequency scanning

To

find the dashed part

of

curves A, C, and

D

we in-vestigate alternatively the dynamic process by frequency scanning.

1.

F)FgD

In the present case we choose

F=0.

100, a value be-tween the critical ones

of

FzD (

=0.

150) and

F„D

(=0.

045) as indicated in Fig. 2. As the frequency co is lower down from 1.3 (path II) with the initial state above curve

B,

the Feigenbaum route to chaos is first seen with onset

of

period two at

co=1.

048 on curve Cand then the boundary crisis occurs at

co=1.

023 on curve

D.

The re-turn map is shown in Fig. 6(a) where the strange attractor contracts suddenly to a fixed point either in the original right valley or in the left one rather than hopping. In

such a case the boundary crisis is referred to as destruc-tive. Figure 6(b) shows the sampled signal

X„as

a func-tion

of

frequency. It demonstrates the destructive feature

FIG.

4. Return maps with

co=1.

2 and (a)

F =0.

194, before crisis; (b)

F=0.

198,crisis occurring oncurve D.

n+1 is due to the nature

of

the symmetrical property

of

the

po-tential well. According to our observations, the state is about equally likely to be in each valley

of

the potential well and give rise to the low-frequency noise with approx-imate

I/f

shape as shown in Fig. 5.

2. co

(

cogD

In this range, curves A, C, and D(dashed part) are not found from this scanning process, because the process is reversible. The hopping state above curve

B

directly re-turns back to the initial stable state once the amplitude

F

is lowered just below curve

B

and vice versa.

(b)

10 Z'. UJ C5

4-O

1.

0-C -2 10 LtJ CL 10 10

06-1.2 I

1.

3 FREQUENCY

FICx. 5. Spectrum of the voltage signal X(t) for

crisis-induced hopping with

F=

0.199and co

=

1.2.

FIG.

6. (a) Return map of boundary crisis with

F=0.

100

and

co=1.

023,(b)response ofvoltage

X„as

afunction ofcowith

with hysteresis after a cascade

of

period doublings. When

F

isset larger than FzD, the crisis with hopping occurs on curve

D.

The map is similar to Fig. 4(b). Hence curve

8

plays an important role in differentiating these two dis-tinct types

of

boundary crisis.

2.

F

&FgD

However,

if

the driving amplitude

F

is chosen between FzD &F &Fztt (0.015),then a hysteresis jump is observed without any bifurcation or crisis occurring. Also, as

F

is less than

Fzz,

the hysteresis disappears.

III.

CONCLUSION

In this paper we have presented asintensive study

of

an analog simulation for nonlinear oscillation with applica-tion to the Duffing equation. Owing to the nonlinear

ef-fect, the dissipative system exhibits the following transi-tion sequences: hysteresis, period doublings, and crises with a global manner. The thresholds are shown in a state

diagram from which the transitions can be easily referred

to.

Two types

of

hysteresis are observed. One is accom-panied by a symmetry breaking

of

the phase portrait as occurred from a symmetrical two-valley to asymmetrical one-valley motion. The other is confined to a one-valley motion only. The former case with the symmetry break-ing appears to be a large size contraction

of

the phase

por-trait.

Two types

of

boundary crisis are also observed. One is associated with a hopping between two strange attractors, and the other is associated with a destruction

of

the

chaotic attractor accompanying a hysteresis jump. The

former is also confirmed to have

1/f

low-frequency noise. The hysteresis upward-jump curve

B

acts as the border-line forthese two distinct types

of

crisis.

Our study not only supports the conjecture

of

Grebogi et al., but also further presents the persistent properties

of

the crisis related to the symmetrical property and the height

of

the central potential barrier in the Duffing equa-tion.

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