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CHAOS 16, 023109 2006

May chaos always be suppressed by parametric perturbations?
Tilo Schwalger
Department of Physics, Humboldt University, Berlin, Germany

Arsen Dzhanoev and Alexander Loskutov
Physics Faculty, Moscow State University, Moscow 119899, Russia

Received 31 October 2005; accepted 23 March 2006; published online 11 May 2006 The problem of chaos suppression by parametric perturbations is considered. Despite the widespread opinion that chaotic behavior may be stabilized by perturbations of any system parameter, we construct a counterexample showing that this is not necessarily the case. In general, chaos suppression means that parametric perturbations should be applied within a set of parameters at which the system has a positive maximal Lyapunov exponent. Analyzing the known DuffingHolmes model by a Melnikov method, we showed that chaotic dynamics cannot be suppressed by harmonic perturbations of a certain parameter, independently from the other parameter values. Thus, to stabilize the behavior of chaotic systems, the perturbation and parameters should be carefully chosen. © 2006 American Institute of Physics. DOI: 10.1063/1.2195787 Great success and profit of the chaos suppression phenomenon in applications led to the widespread opinion that chaotic oscillations may always be stabilized by parametric perturbations. Nevertheless, in what cases can the chaos be suppressed in such a manner? In general, this question means that we should perturb the system within a region of parameter values where chaotic behavior occurs. The chaoticity region may be determined as a set of parameters for which the separatrices are split. In this case, the maximal Lyapunov exponent is always positive. Thus, the chaos suppression implies that all perturbed parameters should not fall outside the limits of this region. In the present paper, by a DuffingHolmes system we construct an analytic example when parametric perturbations cannot lead to the suppression of chaos if they belong to the chaoticity region. One can expect that the same results may be observed in the other physical systems. Our analysis is based on the Melnikov method, which gives us a criterion for the observation of chaos. The obtained results are in excellent agreement with numerical simulations.
I. INTRODUCTION

As is known, chaotic oscillations are inherent in many natural processes. But the development of chaos is sometimes not desirable, and thus it is necessary to create conditions under which originally chaotic systems acquire regular dynamics. In this connection, in the past 15 years, problems related to suppressing and controlling chaos have attracted much attention. Stabilization of chaotic oscillations may be realized in two ways. The first does not take into account the current value of the dynamical variables of the system, whereas the qualitatively different second method involves feedback as a necessary component of the system. By the established convention, the first method is called suppression of chaos and the second one is called feedback controlling chaos.
1054-1500/2006/16 2 /023109/7/$23.00

Using the feedback we have a certain advantage, because in most cases feedback control produces the required result: a prescribed saddle limit cycle is stabilized and the system thus attains the necessary motion. On the other hand, stabilization methods without feedback do not require continuous tracking of the system state and are more robust to noise. The idea that chaos may be suppressed goes back to Refs. 1 and 2, where it was proposed to perturb periodically the system parameters. Later, this approach gained analytical substantiation in a series of publications3­9 as a review, see Ref. 10 . At the same time, it has been found that chaotic systems can be controlled.11­13 We will focus on chaos suppression. This phenomenon means the following. Let us suppose that for required values of parameters, the chaotic dynamics is not acceptable. Then, to realize periodic behavior and suppress chaos, it is sufficient to apply a weak parametric perturbation that does not fall outside the set of parameter values for which the system exhibits chaotic properties, e.g., one that has a positive Lyapunov exponent. Great successes of chaos suppression phenomena in applications led to the opinion that chaos can always be suppressed by such perturbations. However, this is not the case. Using a Duffing oscillator, we analytically construct a counterexample in which periodic perturbations cannot lead to the suppression of chaos. As a consequence, for an arbitrary system we do not know in advance what parameter is appropriate for the stabilization of the system dynamics. The Duffing system is known as one of the simplest nonlinear dissipative models with a wide range of complex behavior. It is used for the description of many real processes, such as mechanical and radio physical oscillations see, e.g., Refs. 14 and 15 and references therein , plasma dynamics see Ref. 16 , and others. The properties of the Duffing system have been widely analyzed. In particular, using the Melnikov method,14,17,18 which is based on the analy© 2006 American Institute of Physics

16, 023109-1

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Chaos 16, 023109 2006

sis of the homoclinical structure of stable and unstable manifolds of a hyperbolic point, it has been shown that at certain parameter values this equation has chaotic dynamics.14,21 Modifications of the Duffing oscillator have also attracted a considerable amount of interest as an appropriate model for the investigation of the chaos suppression phenomenon see Refs. 3, 8, 19, and 20 and references cited therein . One of the modified Duffing equations is known as the Duffing-Holmes system: ¨ x+ x-x+ 1+ cos t x3 = cos t . 1

In the present paper, by means of the Melnikov method we show that periodic perturbations of the parameter do not lead to a stabilization of the chaotic behavior of the system 1 if these perturbations belong to the chaotic region. In the case of the nonlinear oscillatory circuit, perturbations of may be realized by temporal variations of the inductance.

FIG. 1. PoincarÈ section t = const mod T of the system 2 for =0 a and 0 b­d.

II. CHAOTIC REGIONS IN THE DUFFING-HOLMES SYSTEM A. The Melnikov method

It is well known that in Hamiltonian systems, separatrices can split. In this case, stable and unstable manifolds of a hyperbolic point do not coincide, but intersect with each other in an infinite number of homoclinical points. Usually the motion in the n +1 -dimensional phase space x1 , ... , xn , t is considered in the projection onto an n-dimensional hypersurface t = const PoincarÈ section . The presence of such points gives us a criterion for the observation of chaos. This criterion can conveniently be obtained by the Melnikov function MF , which "measures" in the first order of a small perturbation parameter the distance between stable and unstable manifolds. Let us consider a twodimensional autonomous system under the action of a periodic perturbation, x = f0 x + f1 x, t . 2

Let us consider the Duffing-Holmes equation in the following form: x = y, y=x- 1+ cos t x3 - y + cos t . 4

Following Eq. 2 , f0 = y , x - x3 and f1 = (0, cos t cos t x3 - y ). If parameters , , and are small - enough, we can apply the Melnikov method. The MF given by 3 yields
+

D t0 =-
-

cos t -
0

cos

t x3 t - t 0

0

- y0 t - t

y 0 t - t0 dt .

5

Let furthermore x0 be the separatrix of the unperturbed sys tem x = f0 x . Then the MF at any given time t0 is defined as follows:
+

The explicit expressions for the unperturbed separatrix (x0 t , y 0 t ) are determined by the equation H0(x0 t , y 0 t ) 1 1 = 0, where H0 x , y = 2 y 2 - 2 x2 + /4x4 is the Hamiltonian of the unperturbed system. Thus, we obtain x0 t = / cosh t and y 0 t =- sinh t / cosh2 t as a parametrization of the separatrix, where = 2/ . Finally, the MF is22 D t0 =- A sin t0 + B sin with the constants t0 + C 6

D t0 =

-
-

f0 û f

1 x=x0 t-t0

dt ,

3

A= B = 1/24 C = 2/3

sech
4

/2 ,
2

where the integral is taken along the unperturbed separatrix x0 t - t0 and the integrand is f0 û f1 = f 0x f 1y - f 0y f 1x. In general, in dissipative systems one can observe three possibilities for the MF: either D t0 0 Fig. 1 b , D t0 0 Fig. 1 c for any t0, or D t0 changes its sign for some t0 Fig. 1 d . Only in the last case does chaotic dynamics arise. Thus, the MF determines the character of the motion near the separatrix. Note that the Melnikov method has a perturbative character, thus its application is allowed only for trajectories that are sufficiently close to the unperturbed separatrix. Moreover, this method is adaptive only for systems with 1.

4+

2

csch

/2 ,

2

.

Let us now introduce the subset c of the parameter range such that if c, then the system 4 exhibits chaotic properties. To find this chaotic region, we have to examine under which values of the MF changes its sign. For the case = , the analysis gives an explicit expression for c,
cr

3 B-A . 22

7

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May chaos always be suppressed?

Chaos 16, 023109 2006

III. TEMPORAL VARIATION OF THE PARAMETER

Having specified c, let us introduce a parametric perturbation of . We will show that for reasonable amplitudes of perturbation, the chaotic motion of the system 4 cannot be transformed into the regular oscillations. To be more precise, we consider the following equation: x = y, 10 y=x- 1+ cos t x3 -
0

+

1

f t y+

cos t ,

FIG. 2. The maximal Lyapunov exponent. The parameter values are the following: = = 1.1, = 0.114, = 0.03, = 8. Initial conditions: x0 =0, y 0 =0.

If , it has to be considered whether the frequencies and are commensurable. When they are incommensurable, i.e., / is irrational, we immediately have
cr

3 A+B . 22

8

If / = m / n, where m and n are integers, we can find a general expression for cr so that each case of m / n should be analyzed separately. It is clear, however, that commensurability of the frequencies lies between the cases " = " and "m / n is irrational." Therefore, from 7 and 8 we get the critical value of cr, 3 A-B 22
cr

where we made the replacement t = 0 + 1 f t with a normalized periodical function f t = f t + T and amplitude 1. Let us further assume that perturbations should remain within the chaotic region, i.e., 0 c = 0, cr , 0+ 1 0. Below we will show that in this case cr, and 0 - 1 the system 10 does not possess regular dynamics. We will use the Melnikov method, which gives conditions for the appearance of homoclinic points and thus the chaos . Therefore, we will demand the absence of homoclinic points as a condition that the chaos is suppressed. The following numerical analysis will fortify our proceedings. Let us assume that there exists a function f t and an amplitude 1 such that it stabilizes our system 10 . Then D t0 does not change its sign. We will show that this assumption cannot hold together with the requirement that the perturbation lies inside the chaotic region, i.e., t c for all t.

A. Harmonic perturbation

3 A+B . 22

9

For each cr the region of chaos is then given by c = 0, cr . We emphasize that c defines the value set for which homoclinic points exist. This is only a local criterion of the development of chaos in a small enough neighborhood of the unperturbed separatrix.
A. Numerical analysis

In comparison with the analysis described in Sec. II, now has an additional term, 1 f t . Hence, we have to add to D a new part D1 t0 : D t0 = D0 t0 + D1 t0 . Here D0 is the already known MF, D0 t0 =-A sin t0 + B sin t0 + C2 0 , and D1 is determined by the additional part 1 f t :
+

D1 t0 =
-

1

f t y 2 t - t0 dt . 0

11

We numerically analyzed the Duffing-Holmes equation for the following fixed parameters: =8, = 0.114, = 0.03, = = 1.1, and is free. For these values, the expression 7 gives immediately cr = 0.3798. As a criterion of the chaoticity, we calculated the maximal Lyapunov exponent as a function of .23 A typical dependence of the maximal Lyapunov exponent on the parameter is shown in Fig. 2. Positive values mark clearly the region of chaos, c = 0.137 , 0.378 . The same result takes place for any initial conditions chosen in a small separatrix neighborhood. The right boundary value of this region is in excellent agreement with cr found analytically. On the contrary, the left boundary value of c cannot be obtained by the Melnikov method.

Because 0 c, we know On the other hand, to eliminate D t0 does not change its sign. case that the sum D0 t0 + D1 t0 all times t0, i.e., D1 t
0

that D0 t0 changes its sign. chaos we should demand that Let us therefore consider the has a non-negative value for

- D0 t0 .

12

In order to obtain D1, we will analyze f t in the form of a harmonic function f t = sin t + . Then from 11 we have D1 t0 =
1 2

sin t0 +
2

13 /2 .

= /6 2- csch /2 sech with the constant Thus, the expression 12 can now be rewritten as
1 2

sin t0 +

A sin t0 - B sin

t0 - C2 .

14

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Chaos 16, 023109 2006

Let us assume for a moment that B = 0. This simplifies the last inequality containing three frequencies to an inequality with two frequencies. In this case, we can use the following suppression theorem:7 1. 2. To satisfy inequality 14 , it is necessary that = p , where p is an integer resonance condition . If additionally p = 4m +1-2 / / 4n +1 for some integers m and n, then 14 is satisfied only when
min 1 max

,

min

= 1-

C2 R, A 15

max

R = 2, p

/2 . with R =3 2 / p 2- p2 2 sinh p /2 / cosh 2; at p 2 one has to This holds only for p change the roles of min and max. This means that the value set 1 leading to the chaos suppression crucially depends on the order of the resonance p. Now, it is necessary to estimate to what extent we can use the resonant perturbations of different order. To clarify this, let us consider what consequences are implied by the requirement min max. Easy transformations result in
0

FIG. 3. The set S1, where we have the stabilizing of chaotic dynamics, and the set S2 of pairs 0 , 1 , satisfying the condition 0 + 1 cr. One can see that these regions are disjoint.

1

3 C1 - 22

0

.

19

1-

1 p2

cr

.

16

Because 0 1 this inequality cr, for higher harmonics p shrinks drastically the region of perturbations. It is evident that it remains valid if in 14 we take B 0. That is why we confine ourselves by the first-order resonance p = 1. In other words, following the suppression theorem,7 it is necessary to use perturbation of the form sin p + . On the other hand in agreement with the Melnikov approach , if p 1 we can apply such a perturbation only in a very small region on the edge of the chaotic region, i.e., only when 0 cr. Therefore, for an arbitrary 0 in the region of chaos 0, cr we have to take p = 1. Then inequality 16 gives 0 0 cr, which yields the whole chaos region c. The optimal value of the initial phase for suppressing chaos is = 0. This has to be understood in the sense that this value allows the smallest amplitudes of the chaossuppressing perturbation. Thus, together with the resonance condition = = , from 14 we get C1 +
1 2

Thus, this inequality induces a set S2 of "allowed" pairs 0, 1 . The set S2 is also shown in Fig. 3. From this figure we can clearly see that the regions S1 and S2 are not intersecting with each other. This means that there are no pairs 0 , 1 of perturbations within the region c, satisfying the stabilizing condition. Let us therefore show that for any * , 11 S1 and 0 2 * , 12 S2 we have 11 . To make this, it is suffi0 1 cient to show that at * = 0 the maximal possible value of 0 2 1 * 1 is less than 1 and that at 0 = c the minimal possible 1 1 value of 1 as well as 1 is equal to zero. At * =0 see 19 we get 12 =3/2 2 C1 . On the other 0 hand, from 18 , 11 3/2 2 C1 = 12 because max
0
2 3

.

20

sin t0 + C

2

0,

17

= where C1 = B - A, C2 =2 2 0 /3, = . One can show that this inequality is equivalent to the following:
1

This last fact is obtained from the analysis = /6 2 /2 sech /2 . Thus, the maximal value is - 2 csch 2/3 at =0. Consider now some 11 S1 and 12 S2 at * = cr 0 2 =3/2 2 C1 . It is obvious that 1 = 0. According to inequality 18 at * = cr the boundaries of S1 are determined as 0 follows:
1 1

+

C

1 2

2 3
0

.

18

+

C

1 2

=

2 3
cr

=

C

1 2

.

Then the solution set for 1 turns out to be directly visible. In Fig. 3 the set S1 of pairs 0 , 1 for which the chaotic motion vanishes i.e., D t0 does not change its sign is shown. However, we consider perturbations of that should remain within the chaotic region 0, cr . Therefore, 0 + 1 2 C1 or cr =3/2

Therefore, the minimal value of 11 is also equal to zero. Thus, Fig. 3 correctly illustrates the relation of S1 and S2, i.e., S1 S2 = . It should be noted that in order to find the regular dynamics, we have analyzed the inequality D t0 0 for all t0. Following to the Melnikov method, however, it is sufficient

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May chaos always be suppressed?

Chaos 16, 023109 2006

summand D1 t0 in the MF. We will require again that by this part the Melnikov function D t0 does not change its sign. In particular, analogously to 12 , D1 t
0

- D0 t0 = A sin t0 - B sin

t0 - C

2

21

FIG. 4. The maximal Lyapunov exponent for the system 10 . Each negative value is marked with a black dot. The solid line marks the upper boundary of S2. The dashed line represents the boundary of S1 obtained by the Melnikov method. The parameter values are the following: = = 1.1, = 0.114, = 0.03, = 8. Initial conditions: x0 =0, y 0 =0.

for all t0, where D0 t0 changes sign in the analyzed chaotic region. If we arrange f t as a sum of harmonic functions with different frequencies, then, according to 13 , we obtain that the left-hand side of the last inequality can also be represented by a sum of harmonic functions with the same frequencies. According to the suppression theorem see Sec. III A , inequality 21 can be satisfied if both sides consist of the same frequencies considered as before only in the first-order resonance. That is why we should take the perturbation as follows: f t = a1 sin t + a2 sin t, 22

where a1 and a2 are coefficients satisfying the relation a1 + a2 =1 . 23

that D t0 does not change its sign, which obviously also includes the case D t0 0 for all times t0. Then the relation 17 may be changed by C1 + 1 2 sin t0 + C2 0. But, because C2 is a positive value, this inequality cannot be carried out. Thus, we come to the conclusion that in the DuffingHolmes system the suppression of chaos cannot be achieved by a harmonic perturbation of the parameter within the region c.
1. Numerical analysis

Thus, the correcting term in the MF is D1 t0 = 1 2 a1 sin t0 + a2 sin t0 . Therefore, inequality 21 yields
1 2

a

1

- A sin t0 +

1

2

a

2

+ B sin

t0 + C

2

0. 24

To analyze the perturbed system numerically, we computed the maximal Lyapunov exponent for different pairs 0 , 1 . For convenience, we have to norm 1 to 1 / 0. The result of 250 250 calculations of 0 , 1 / 0 , 0 c, is shown in Fig. 4. One can see that our theoretical predictions are in fine agreement with the numerical analysis. In the region S2 the solid line we have everywhere a positive value of the Lyapunov exponent. Therefore, here the perturbations are not stabilizing for the system. In the region S1 the dashed line , which was obtained by the Melnikov method, we clearly see the stabilization phenomenon.
B. Perturbations with frequencies å

The chaos suppression phenomenon is equivalent to the satisfaction of this inequality for some a1, a2, and 1 0. Because the factors before sin should be as small as possible we remind the reader that C2 0 , without loss of generality 0 and a2 0. we can require that a1 Furthermore, according to our strategy, perturbations should belong entirely to c. This region can be exactly specified in the particular cases " = " and " / is irrational" see 7 and 8 . Hence, for perturbations with two different frequencies, the inequality 24 may be considered are incommensurable. Below we will focus when and on this case. First of all, note that the inequality 24 is equivalent to the following: min
t0 R 1 2

a

1

- A sin t0 + 0.

1

2

a

2

+ B sin

t

0

+C

2

In the previous section a parametric perturbation with only one frequency was analyzed. As a consequence, to eliminate homoclinic points we had to put = . Let us now . The subharmonic resonances are not suppose that considered for the same reason as in Sec. III A: If we allowed higher-order resonances, the chaos suppression would not be possible in almost the whole region of chaos of the parameter 0. Only in a very small region near cr could one satisfy the suppression theorem. This does not yet mean that there exist chaos suppression perturbations that are strictly in the region of chaos. In this case, it is necessary to vary f t , which induces see the previous section an additional

Owing to the incommensurability of and , we can evaluate separately the minimal value of every summand. Thus, from 24 we have
2

a

1

1

-A +

2

a

2

1

-B

C2 .

25

This inequality can be easily analyzed geometrically Fig. 5 . In this figure, both summands of inequality 25 are shown by thin lines, and the total sum on the left-hand side is shown by the bold line. Considering the right-hand side, C2 0, we can find the set S1 of inequality 25 . Because this inequality is symmetrical with respect to both summands, without loss of generality we can assume that the zero of the second summand lies to the left of the zero of the A / 2a 1 . first summand: B / 2 a2

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023109-6

Schwalger, Dzhanoev, and Loskutov

Chaos 16, 023109 2006

a

1

-2 a

A +1 a B -2 -a

2

+

2 3
2

1

a1 a2
2

+1 a + + 2 3

+

2 3

=- a -

1

2 a1 + a 3

2

2 =0 . 3

IV. CONCLUSION
FIG. 5. The set S1 of the stabilizing values 1 see inequality 25 for the given C2 0 and the set S2 satisfying the condition that perturbations should belong to the chaotic region cr.

Now let us define S2 as the set of pairs 0 , 1 for which the perturbation remains within the chaotic region c: 0 +1 cr. Then we can show that S1 and S2 are not intersecting with each other, which means that perturbations inside c cannot be found. Consider the values C 1 and C 2 , which are the y coordinate of the points P1 and P2, respectively see Fig. 5 . To find that S1 S2 is the empty set, it is sufficient to obtain that C 1 C 2 , and both the upper bound of S2 and the bold line i.e., the left-hand side of inequality 25 meet at the point 1 = 0. The last condition is obvious. Thus, we should show that C 1 C 2 . The explicit forms of C 1 and C 2 are

C

1

=

2

a

1

2

-B a2

-A =

a1 a2

B+A ,

C

2

=

2 3

2

0

=

2 3

2

cr

-

1

=A+B+

2B . 3 a2

Thus, it is necessary for the inequality

A+B+

2B 3 a2

a1 a2

B+A

26

to be satisfied. First, consider the case when the expression under the module sign is not negative. Then inequality 26 can be rewritten as - a1 - a2 +2/3 0. In fact, this is correct because - a1 - a2 +2/3 -2/3 a1 + a2 +2/3=0. Here we have used the relations 20 and 23 . Let us now consider the case in which the expression under the module sign is negative, i.e., A / B a1 / a2 . Then inequality 26 is equivalent to a1 - 2A / B +1 a2 +2/3 0. This inequality is also right because

The development of the theory of dynamical systems and numerous studies of nonlinear processes has shown that chaotic behavior is typical and prevalent in many nonlinear processes. Nowadays it is obvious that chaotic properties are inherent in the overwhelming majority of systems, and if chaos is not detected, this is perhaps due to the fact that it arises in very small regions of the parametric space or for nonphysical parameter values. The predictability problem, which first arose for fairly complex systems such as hydrodynamic systems or systems in statistical mechanics , became common for many fields of modern science. At the same time, as is known, chaotic dynamical systems are pliable to external perturbations. This property can be used to the control of dynamical systems and suppression of the onset of the undesirable chaotic regime. This means that parametric perturbations should be determined at the set of their values for which the system has a positive maximal Lyapunov exponent. If this is not the case, we cannot assess the chaos suppression. This is the main sense of this phenomenon. However, in the present article we come to the main conclusion that the phenomenon of chaos suppression cannot be achieved via arbitrary parametric excitations. Analyzing the known Duffing-Holmes model 1 , we found that chaotic dynamics cannot be suppressed by a harmonic perturbation of the parameter , independently from the other parameter values. In addition, we have shown in the case in which and are incommensurable that chaos suppression cannot be realized even by an arbitrary periodic perturbation of . On the basis of our analysis, we have come to the conclusion that to stabilize the behavior of a chaotic system, the perturbation and parameters must be carefully chosen. This contradicts many previous expectations that the chaos may always be suppressed parametrically. If, however, we apply short-time impulse perturbations of the parameter , then we can easily stabilize the system dynamics.24 One can expect that similar results will be observed in other systems. In this connection, it would be useful to have a basic criterion that would allow us to determine in what cases the chaotic motion can be stabilized by parametric excitations. This important question will arise every time in practice. However, as of yet there are not even general analytical criteria of the existence of chaos in quite simple dynamical systems.

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023109-7
1

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15

16

17 18

19 20

21 22 23 24

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