Suppressing chaos in the duffing oscillator by impulsive actions

Chaos, Solitons & Fractals, Vol. 9, No. 1/2, pp. 307-321, 1998 ~) 1991~ Elsevier Science Ltd. All rights reserved Printed in Great Britain 0960-0779/98 $19.00 + 0.00

Pergamon

PII: S0960-0779(97)00069.6

Suppressing Chaos in the Dulling Oscillator by Impulsive Actions G. OSIPOVt, L. GLATZ and H. T R O G E R

Technical University Vienna, Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria

Abstract--The method of feedback impulsive suppression of chaos is introduced. Our approach is based on the similarity of return maps of some time continuous systems with one-dimensional cubic maps. The method is illustrated for the Dumng oscillator with positive linear stiffness by making use of its symmetry properties. Q 1998 Elsevier Science Ltd. All rights reserved

1. I N T R O D U C T I O N

Controlling chaotic behavior has b e c o m e an active field in the study of nonlinear dynamics in the last few years (e.g. [1-3]). Algorithms used in controlling chaos can be roughly classified into two groups [2]. One way of controlling chaos, used in [1, 3], is stabilizing an unstable periodic orbit which is e m b e d d e d in a chaotic attractor. In the second group, control is achieved by suppressing chaos. H e r e several different approaches have been proposed in the literature. A n overview can be found in [2]. Most algorithms from this group, in general, use either a parametric or external perturbation which is continuous in time (e.g. [4-6]). As a rule these actions are harmonic. In this paper, we introduce an algorithm of suppressing chaos in continuous dissipative system with an external impulsive force. A necessary condition to use our strategy is a reduction, at least approximately, of continuous flows to time discrete one-dimensional maps. Since the time instant and the duration of the impulse depend on the state of the system, this algorithm is a kind of feedback control. T h e m e t h o d is illustrated with the example of the Duffing oscillator [7], for which other methods of controlling chaos have been used (e.g. [8, 9]).

2. PROPERTIES OF THE D U F F I N G OSCILLATOR WITH POSITIVE LINEAR STIFFNESS

The Duffing equation can be written in the form "~ Og)~ -{'- X q- ~ X 3 = B c o s

oJt,

tPermanent address: Nizhny Novgorod University, Gagarin Ave. 23, 603600 Nizhny Novgorod, Russia. 307

(1)

G. OSIPOV et al.

308

where a , y, B and w are parameters. This equation also describes the motion of a particle in a field with the potential 1

V(x) = ~ x 2 + ~ 7x 4.

(2)

The nonlinear effect of the restoring spring in the Dufling oscillator observed in many mechanical problems is expressed by the cubic term in eqn (1). We investigate both positive and negative nonlinear stiffnesses. In the first case (y < 0), the spring is softening and the corresponding potential in eqn (2) has two maxima and one minimum; in the second case (3' > 0), a hardening spring is given corresponding to a potential with one minimum only. For the numerical calculations, the following p a r a m e t e r values were investigaled: for negative nonlinear stiffness: c~ = 0.4, y . . . . 0.2, B = 1.0,

(3)

for positive nonlinear stiffness: c~ = 0.2, y = 1.0, B = 27.t).

{4)

For further discussion of eqn (1), it is better to write the equation as a first-order system S: = y = v , ( x , y , z ) ,

)) = --(~'y

Y -- "yX3 + B cos Z = v2(x,y,z ),

(5)

~. = ~o = v3(x, y , z ),

on the natural phase space R 2 x S ~. Due to the symmetric potential (see eqn (2)) of the system, the vector field ~(x,y,z) generated by the system (5) has the following properties: v,(x.y,z)

= -v~(-x,-y,z

+ Jr),

o ~ ( x , y , z ) -- - o 2 ( - x , - y , z

+ Jr),

(6)

P3(x,y,7.) = V3(-x,--y,7. + It). This s y m m e t r y of the vector field has the following consequences on the symmetry of trajectories in phase space. There exists either a symmetric orbit S(x(t),y(t),z(t)) with

(x(t),y(t),z(t)) = ( - x ( t ) , - y ( t ) , z ( t )

+ Jr),

(7)

or two asymmetric orbits Si(xl(t),yl(t),z.(t)) and S2(x2(t),y2(t),z2(t)) which are symmetric to each other:

(x1 (O,y, (t), z, (t))

= ( - x ~ ( t ) , - y 2 ( t ) , z~(t) + Jr).

(8)

In [10, 11], it is shown that asymmetric orbits come into existence pairwise. Obviously symmetric periodic orbits can only exist with odd period. Therefore the period-doubling bifurcations that lead to chaos in (5) appear only for asymmetric periodic orbits and only when the s y m m e t r y of the bifurcating periodic orbits has been broken. As a consequence of the symmetry properties in our system, two period-doubling cascades coexist, for both of which the value of the bifurcation p a r a m e t e r s are the same. To understand the bifurcations in the motion of the Duffing oscillator (5) for the

Suppressing chaos in the Duffing oscillator

309

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= 0.85

i 0.855

i 0.86

i 0.865

i 0.87

I 0.875

i 0.88

f 0.885

to 0.89

Fig. 1. Bifurcation diagram of the Duffing oscillator with positive linear stiffness.

parameter values (3), we vary the frequency to of the external force and define a global Poincar6 map P as the map of the (x,y)-plane at z = c = constant onto itself P:Zc~Zo

(x,y)~P(x,y)=frlz, (x,y).

(9)

Here f r is the flow generated by the dynamical system (5) on the phase space R 2 × S', T = 2re~to is the period of the external force and c is a constant determining the location of the Poincar6 plane. N o w we discuss s o m e of the properties of the Duffing oscillator for the parameter set (3), that is, for negative nonlinear stiffness. In Fig. 1, the bifurcation diagram for decreasing driving frequency is shown. The large symmetric periodic orbit which exists for to > 0.884 is broken into two asymmetric stable periodic motions. After the symmetry-breaking bifurcation at to = 0 . 8 8 4 , three period-1 orbits coexist in phase space. At to = 0 . 8 5 8 6 , two period-doubling cascades of the two asymmetric periodic orbits start. This leads to the birth of two mutually symmetric chaotic attractors at to = 0.8538. At to = 0.8508, these chaotic attractors merge into one symmetric chaotic attractor. In the chaotic interval (0.8476,0.8538) narrow windows with periodic motions exist. Decreasing the driving frequency b e l o w 0.8474 all solutions diverge to infinity. We n o w consider return maps x,÷] versus x,, where x , is the x-coordinate of the intersection of the phase trajectory with the Poincar6 plane. In Fig. 2, for c = 0 such return maps at different values of the frequency to are shown. Figure 2a shows two stable periodic orbits at to = 0.86. Figure 2b shows one of the two existing chaotic attractors at to = 0.852. Figure 2c shows the 'big' chaotic attractor at to = 0.848 and Figure 2d the map for the set of

G. OSIPOV et al.

310

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-2.5

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m(nT)

x((n + 1 ) T ) 0

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(c)

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i -0.5

x(nT) 0

Fig. 2. R e t u r n maps corresponding to Fig. 1 f o r c = 0: (a) t w o stable periodic orbits at oJ = 0.85, (b) one o f the t w o chaotic attractors at w =0.852, (c) 'bi~' chaotic attractor at ~o = 0.848, (d) trajectories escapin~ to infinity at = 0.846.

312

G. OSIPOV

et al.

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L

0.5

\

LI

\

02

\

\ \

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\

-0.5

/ -1

-1.5 v -1.5

i -1

i -0.5

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~ an 1.5

Fig. 3. C u b i c m a p L a n d its t w o s h i f t e d b r a n c h e s L~ a n d L 2 f o r A = 0.5.

trajectories which diverge to infinity at o~ = 0.846. The key observation now is that all these return maps have m o r e or less the form of a cubic polynomial. Consequently, iterating the cubic m a p (Fig. 3 curve L f o r / x = 3) x,,+t =/zxn(1 - x,2)

(10)

yields almost the same dynamics as the return maps of the Duffing oscillators shown in Fig. 2. Moreover, for (10) we obtain the bifurcation diagram x(/x) presented in Fig. 4. At # = 1, a pitchfork bifurcation takes place which corresponds to the symmetry-breaking bifurcation for periodic orbits in eqn (1). One stable fixed point xl = 0 gives birth to two stable points x2 = + V ~ - 1 / # and x3 = -X/~-~ - 1//, and the fixed point Xl becomes unstable. At /, = 2, for both fixed points x2 and x3, the first period-doubling bifurcation takes place. Starting here, two period-doubling cascades lead to the birth of a pair of symmetric chaotic orbits at /x = 2.3 whose realization depends on the initial condition x0. After these chaotic orbits meet at tz = 2.65, only one 'big' chaotic orbit exists independent of the initial conditions. It ceases to exist at ~ = 3. For t* > 3, all solutions of (10) diverge to infinity. Our idea of suppressing chaos in the Duffing oscillator is based on two facts: 1. the return maps for the form; 2. there is good qualitative Duffing equation ( I ) for increasing values of the

Duffing oscillator (1) are nearly one-dimensional and have cubic and quantative agreement of the evolution of the solutions of the decreasing driving frequency w and the cubic m a p of eqn (10) for p a r a m e t e r #.

Suppressing chaos in the DuRing oscillator

313

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a

-0.5

-1.5 0

J 0.5

a 1

I 1.5

t 2

J 2.5

3

/z

Fig. 4. Bifurcation diagram x(/.) of the map (10).

At first, we will apply the method of impulsive chaos suppression to the cubic map (10) and then we implement this method for the Duffing oscillator.

3. SUPPRESSION OF CHAOS IN A CUBIC MAP

In general, parametric control of return maps which generate chaotic motions consists in variable or parameter perturbations such that the perturbed map produces a regular motion. Many m e t h o d s of controlling chaos have been developed for one-dimensional maps. Perhaps the simplest way to suppress chaos has been proposed in [12]. It is shown that by applying a constant signal at any iteration of an unimodal map chaotic behavior can be suppressed to obtain any desired periodic attractor. Following [12], we add the constant signal A at each iteration step to the right-hand side of eqn (10) for chaos suppression in the one-dimensional cubic map (10). This gives in x.+, = A +/*x.(1 - x2).

(11)

The sign of a is chosen to be positive for x . < 0 and negative for x. > 0. Hence we have A0, a=t_)t0,

i f x , < 0, ifx,>0,

314

G. O S I P O V et al.

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-0.45

-0.4

-0,35

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-0.15

-0,1

-0.05

Fig. 5. Bifurcation diagram x.(A) of the cubic map ( l l ) with constant feedback at x~ = 0.7.

/x =

-

0

2.75 with initial condition

with Ao>0. Adding A to (10) means that at each iteration the value of x,+, decreases or increases by the same constant value. In the graph of the m a p (see Fig. 3), we have denoted the left part (x < 0 ) of curve (10) by L, and the right part (x > 0 ) by L 2. Hence, for increasing values Ao, the curve L~ moves up and the curve L2 moves down. The graphs in Fig. 3 show the curves L1 and L2 for Ao = 0.5. Obviously such a shift leads to a change of attractors in (10). For example, for Ao = 0.5, two stable fixed points O~ and 02 exist. Let the value of the p a r a m e t e r t~ be in the interval (2.3,2.65) (see Fig. 4). Depending on the initial conditions (xo > 0 or x0 < 0), one of two chaotic motions will be obtained either for x > 0 or x < 0. For Xo > 0, a chaotic sequence x, exists such that all x,1 > 0. Therefore this sequence is completely represented by L 2. Hence this case, in principle, is the same as treated in [10]. We now iterate the cubic m a p of eqn (11) at/~ = 2.75 in the fully chaotic domain. Figure 5 shows the bifurcation diagram x,(A) for xo = 0.7. It can be seen that the system behavior for decreasing values of Ao is similar to that for increasing values of /~ (Fig. 4). Chaos suppression can now be found for a wide range of A values. For example, for A = -0.15, we can suppress chaos and convert the chaotic behaviour into a period-3 motion. Note that, if A < - 0 . 5 4 , the trajectories diverge to infinity. In addition to the method proposed in [12], we have also applied some modification. We have changed only one part of the function of the map. For example, if we shift down the curve L (Fig. 3) only for values x,, < - 0.8, suppression of chaos is also possible.

Suppressing chaos in the Dulling oscillator

6

315

I

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1

5

:'

Eo=5

..

4

:

3

2

Eo = 2

Eo 0 -0.5

0

0.5

i

-n

1

1.5

Eo

0.5 =

0.1

t 2

Fig. 6. Response of the linear (solid lines) and nonlinear (dashed lines) Duffing oscillator to an impulsive excitation. 4. RESPONSE OF THE DUFFING OSCILLATOR TO AN IMPULSIVE SIGNAL

Now we show that the same effect obtained for the point m a p by adding a constant signal can be achieved for the Duffing oscillator by an impulsive function E(t) acting on the right-hand side of eqn (1). That is, we consider + cr~ + x + 3"x3 = B cos tot + E(t),

(12)

where

E(t)

SEo, I L 0,

if t e A t , if t ft At.

H e r e At is the interval of the impulsive action and Eo is the magnitude of the impulse. For the further analysis, we calculate the response of the Duffing oscillator to the impulsive force E ( t ) only for the linear part of the system (1), that is for 3' = 0. The justification for this simplification is shown in Fig. 6, where a comparison of the response of the linear oscillator (solid lines) given by the formulas below and the nonlinear oscillator (dashed lines) calculated by numerical integration under the impulsive action are presented. The impulsive force is applied at t = 0. For At ~ (0,0.5), there is excellent agreement for the linear and nonlinear oscillators. In fact, the agreement in this time interval holds up to values as high as Eo = 50.

316

G. O S I P O V et al.

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[

I

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i

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-45

-40

-35

-30

-25

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0

Fig. 7. B i f u r c a t i o n d i a g r a m of x(E~0 at o) = 0.852 c o m p u t e d from the P o i n c a r 6 m a p at c = 0.

We denote the deviation of the variables x and y due to the impulse by ~ : and Ay, respectively. For the linear system (y = 0 in (1)), Ax and Ay can be given explicitly by

(13) O) d

Ay = E~ (e_¢, sin LOd

wdt)U(t),

(14)

where Wd = V1 - a2/4 is the frequency of the d a m p e d oscillation and ~: = a/2 is the damping factor. Also U(t) is the unit step function defined by U(t) = 0 if t < 0, U(t) = 1 if t -> 0. From the expressions in eqns (13) and (14), it follows that the signs of Ax and 2~y are the same as that of E0. We now apply the impulsive signal for the chaos suppression to the Duffing equation (1) for the two values: w = 0.852 and 0.848. In the first case, two symmetric chaotic orbits coexist in phase space. In the other case, only one symmetric chaotic motion exists. We start with the suppression of the chaotic motion at ~o = 0.852. From the Poincar6 map calculated at c = 0 , the bifurcation diagram x(Eo) shown in Fig. 7 is obtained. The x-coordinate of the chaotic attractor is drawn versus the magnitude E0 of the m o m e n t a r y impulse (in eqn (12), E(t) = G~6(t)). F r o m Fig. 7, it can be seen that the chaotic motion can

Suppressing chaos in the Duffing oscillator

317

be converted into periodic motions with different periods, depending on the choice of Eo. For example, at E0 = - 2 0 a period-4 motion, at Eo = - 3 0 a period-2 motion, and at Eo = - 5 0 a period-1 motion will be obtained. We note that in all these cases the chaos suppression takes place only if the magnitude of Eo is rather large. A conversion into periodic motion, however, is also possible at smaller values of Eo by increasing the interval of duration At of the action E(t). The time interval At can be considered as the union of the time intervals of the impulsive actions

At = (t~,t~) U"" U (t~'l,g),

(15)

where N is the number of intervals. The difference b e t w e e n two neighbouring intervals is constant and equal to the period of the external driving force, that is, ti ÷1 - t~ = T = 2zr/to. Results of chaos suppression in the case when the impulse acts in the z-interval zl = 0, z2 = 0.2 are presented in Fig. 8, and for the z-interval zl = 0, z2 = 0.05 in Fig. 9. For example, from the bifurcation diagram in Fig. 9, it is clear that at Eo = - 0 . 0 8 8 5 a period-6 m o t i o n is obtained. In Fig. 10, the chaos suppression for the parameter values of Fig. 9 and Eo = - 0 . 0 8 8 5 is shown. The impulsive action is applied in the t-interval tl = 75T ~ 5 5 3 . 1 , t2 = 2 0 0 T ~ 1659.3. Next we consider to = 0.848. In this case, either a chaotic attractor or divergence of trajectories to infinity can occur. According to the results presented in Fig. 2c, a period-1 unstable motion in phase space intersects the Poincar6 plane at c = 0 in the point O, where x* -~ - 1 . 5 , y* ~ - 1 . 5 . The value x* is the boundary value such that if x < x * then we apply

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Fig. 8. Chaos suppression for an impulse acting in the z-interval zl = O, z2 = 0.2, at ~ = 0.852.

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i -0.25

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Fig. 9. C h a o s s u p p r e s s i o n for an i m p u l s e acting in the z - i n t e r v a l z] = 0, z2 = 0.05, at w = 0.852.

z(~,.T) -0.85

-0.9

-0.95

-1

-1.05

-1.1

-1.15

-1.2

-1.25

-1.3 200

i 400

i 600

i 800

t 1000

t 1200

I 1400

I 1600

i 1800

t 2000

Fig. 10. T i m e h i s t o r y of c h a o s s u p p r e s s i o n for an i m p u l s e acting in the z - i n t e r v a l z] = 0, z: = 0.05 with an a m p l i t u d e E~) = 0.0885 y i e l d i n g a period-6 m o t i o n at ~o = l).852.

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Chaos suppression f o r an impulse acting in the z - i n t e r v a l zl = 0, zz = 0.2, at eo = 0.848.

=((n + 1)T) 4.6

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4

4.2

4.4

x(nT) 4.6

Fig. 12. R e t u r n m a p of o n e of the two c h a o t i c a t t r a c t o r s of the Duffing o s c i l l a t o r w i t h p o s i t i v e n o n l i n e a r stiffness at = 1.43.

320

G. OSIPOV

et al.

x(n:r) 0.5,

1

I

I

I

I

:: I ilt t111tltit ¸ i ti t i

-0.1

500

I

J

I

J

i

600

700

800

900

1000

1100

Fig. 13. Time history of chaos suppression for an impulse acting in the z-interval z I 0, ~'2 0.5, with an amplitude E~)= 2 yielding a period-4 motion at w - 1.43. :

:

a positive impulse E0 and if x > x * we apply a negative one. In Fig. 11, the result of suppression of chaos with the impulse acting in the z-interval z~ = 0, z2 = 0.2 is presented. Finally it should be m e n t i o n e d that even in the region where all trajectories for arbitrary initial conditions escape to infinity (e.g. co = 0.846), a transformation into a periodic m o t i o n is possible. T h e a p p r o a c h is the same as in chaos suppression but has to start immediately. So far we have considered the Duffing oscillator with negative nonlinear stiffness. If a positive nonlinear stiffness term is introduced, no trajectories escape to infinity following f r o m the f o r m of the potential in eqn (2). For the p a r a m e t e r values (4) and at w - 1.43, two chaotic attractors exist in phase space. T h e return maps of one of these attractors is shown in Fig. 12. T h e suppression of the chaotic m o t i o n was achieved by an impulsive action with m a g n i t u d e E0 = 2 acting during the z-interval zj = 0, z2 = 0.5. The result is the period-4 m o t i o n shown Fig. 13. Thus the a p p r o a c h of suppressing chaos is identical for negative and positive nonlinear stiffness.

5. CONCLUSIONS W e have shown that it is possible to convert the chaotic attractor of the Duffing oscillator with positive linear and b o t h positive and negative nonlinear stiffness into a periodic attractor. This is d o n e by applying an external impulsive action with a period equal to the period of the external periodic force. T h e a p p r o x i m a t e value of the m a g n i t u d e and the length of the impulse can be o b t a i n e d analytically. M o r e o v e r , this m e t h o d alh)ws us to o b t a i n a stable periodic (limited) m o t i o n even if only diverging trajectories exist.

Suppressing chaos in the Duffing oscillator

321

Our method of suppressing chaos can also be realized if impulsive parametric perturbations are used. For example, for the system + Ot.~ + X + ('y + E ( t ) ) x 3 = B c o s ogt,

(16)

where E(t) is the impulsive action, the chaos suppression is possible. Detailed analysis of this case we expect to present in the future. Finally we note that the proposed method of chaos suppression can be applied to all systems where a period-doubling cascade is observed. The application to autonomous systems, however, deserves special care. For example, for three-dimensional systems of ordinary differential equations, like Chua's circuit [13] or R6ssler's equations [14], one will have to select a suitable Poincar6 plane. If a trajectory intersects this plane, an impulsive action should be applied. Obviously the impulsive action in this case will not be periodic anymore. Acknowledgements--The first author thanks the Institut fur Mechanik of the Technical University of Vienna for its hospitality and Osterreichischer Akademischer Austauschdienst and RFBI (grant 96-02-18041) for their financial support. In addition, support by the Austrian Science Fonds (FWF) under grant P10705-MAT for the second author is gratefully acknowledged.

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