─юъєьхэЄ тч Є шч ъ¤°р яюшёъютющ ьр°шэ√. └фЁхё юЁшушэры№эюую фюъєьхэЄр : http://chaos.phys.msu.ru/loskutov/PDF/JETP5_04Los_Dzh.pdf
─рЄр шчьхэхэш : Mon Jan 31 16:05:56 2011
─рЄр шэфхъёшЁютрэш : Mon Oct 1 19:59:53 2012
╩юфшЁютър:

Journal of Experimental and Theoretical Physics, Vol. 98, No. 5, 2004, pp. 1045н1053. Translated from Zhurnal шksperimental'nooe i Teoreticheskooe Fiziki, Vol. 125, No. 5, 2004, pp. 1194н1203. Original Russian Text Copyright й 2004 by Loskutov, Dzhanoev.

NONLINEAR PHYSICS

Suppression of Chaos in the Vicinity of a Separatrix
A. Yu. Loskutov and A. R. Dzhanoev
Moscow State University, Moscow, 119899 Russia e-mail: loskutov@moldyn.msu.su
Received April 30, 2003

Abstract--The standard Melnikov method for analyzing the onset of chaos in the vicinity of a separatrix is used to explore the possibility of suppression of chaos of dynamical systems of a certain class. Analytical expressions are obtained for external perturbations that eliminate chaotic behavior. These results are supplemented with a numerical analysis of the DuffingнHolmes-oscillator and pendulum equations. й 2004 MAIK "Nauka / Interperiodica".

1. INTRODUCTION Intensive theoretical and experimental studies of chaotic dynamical systems revealed their unexpected and remarkable property: they are highly susceptible and extremely sensitive to perturbations. This discovery served as a starting point for finding a means to control the behavior of chaotic systems, i.e., to change from chaotic regimes to required regular oscillatory regimes by means of relatively weak perturbations. Suppression of unstable or chaotic behavior of dynamical systems is generally achieved via stimulated excitation of stable (usually periodic) oscillations by means of multiplicative or additive perturbations. In other words, an external perturbation is required to change from a chaotic state of a system to a regular regime. The statement of the problem is outwardly simple, but its solution is very difficult to find for particular dynamical systems. Moreover, even though the problem has been analyzed in numerous studies, a systematic and rigorous theory of suppression of chaotic behavior has been developed only for some common families of dynamical systems (see [1, 2] and references cited therein). Chaotic behavior can be suppressed by two different methods. In one of these, the state of a system is changed from chaotic to regular by perturbation without feedback. In other words, this method does not make use of the current values of dynamic variables. In the other method, the perturbation is adjusted in accordance with the required values of dynamic variables; i.e., feedback is an integral component of the dynamical system. By convention, the former method is called open-loop suppression (or control) of chaotic dynamics. The latter method is called feedback control of chaotic systems. Both methods can be implemented either parametrically or by direct forcing. To the best of our knowledge, the first analyses of suppression of chaotic dynamics of certain systems

were presented in [3, 4]. However, extensive research along these lines was initiated by [5, 6], where it was shown that relatively weak parametric perturbations can be used to regularize a particular saddle orbit embedded in a chaotic attractor. These and other results stimulated studies of suppression of chaotic dynamics and evoked great interest in control of unstable systems. A vast number of numerical and experimental studies were focused on the possibility of suppression of chaos and implementation of periodic or other required dynamics in various systems and maps (see [1, 2, 7н10] and references therein). The standard Melnikov method is an effective tool used in analytical treatments of the problem of chaos suppression [11]. It is based on comparison of the firstorder terms in the series expansions of the solution in terms of a perturbation parameter on stable and unstable separatrices. In particular, the Melnikov method was applied to explore the possibility of elimination of chaotic dynamics of the DuffingнHolmes oscillator [12н16] (see also [17]). It was shown that a small parametric perturbation of the system's chaotic dynamics suppresses chaos. Furthermore, the Melnikov method was used in [18] to examine the effects of parametric perturbations in a model of the Josephson junction. In this paper, the Melnikov method [11, 19] is applied to find analytical expressions for parametric perturbations that suppress chaotic and/or unstable behavior of dissipative dynamical systems. The DuffingнHolmes oscillator and pendulum are considered as examples. 2. THE MELNIKOV METHOD In this section, we briefly describe the Melnikov analytical method for identifying homoclinic or heteroclinic chaos, relying on the original paper [11] (see also [19н21]).

1063-7761/04/9805-1045$26.00 й 2004 MAIK "Nauka / Interperiodica"

1046
u x0 s X0 x0 (a)

LOSKUTOV, DZHANOEV

Omitting intermediate calculations, we write out an expression for D:

(b)

D ( t, t 0 ) = н

н

f 0 f 1 dt .

(2)

(c)

(d)
Fig. 1. Split separatrix loops.

This function determines conditions for chaotic behavior of the original system. In the domain where the sign of D(t, t0) alternates, the separatrices intersect and the system exhibits chaotic dynamics. 3. ELIMINATION OF CHAOTIC DYNAMICS IN THE VICINITY OF A SEPARATRIX We use the mathematical procedure described above to explore the possibility of suppressing chaotic dynamics for systems with separatrix loops described by Eq. (1), where f 0 ( x ) = ( f 01 ( x ), f 02 ( x ) ) , f 1 ( x ) = ( f 11 ( x, t ), f 21 ( x, t ) ) . (3)

Consider a simple autonomous system with a single hyperbolic point X0 subject to a periodic perturbation: x = f 0 ( x ) + f 1 ( x, t ) , (1)

where x = (x1, x2) and f1 is a periodic function with period T. Suppose that the unperturbed system (with = 0) has a single separatrix x0(t) (see Fig. 1a):
t ▒

lim x 0 ( t ) = X 0 .

For such a system, the Melnikov function D(t, t0) can be written as

The separatrix is split by the perturbation, i.e., has distinct incoming and outgoing branches. Three possibilities arise as a result: the separatrices either do not intersect (in which case one may enclose the other, see Figs. 1b and 1c) or intersect at an infinite number of homoclinic points. Chaotic dynamics are observed only in the latter case (see Fig. 1d). To find an intersection condition, one must use a perturbation method to calculate the distance D(t, t0) between the separatrices at an instant t0 . If the outgoing separatrix encloses the incoming one, then D(t, t0) < 0. If the incoming separatrix encloses the outgoing one, then D(t, t0) > 0. Only if there exists t0 such that the separatrices intersect, then the sign of D(t, t0) alternates. In the method substantiated in [11], the distance D(t, t0) between the branches of a split separatrix is determined by performing integration along unperturbed trajectories. The method is based on comparison of the first-order terms in the series expansions of the solution in terms of the perturbation parameter on stable and unstable separatrices. To calculate D(t, t0), it is sufficient to find the solutions on the stable and unstable manifolds, xs and xu. When = 1, these solutions differ by the vector r ( t, t 0 ) = x ( t, t 0 ) н x ( t, t 0 ) = x ( t, t 0 ) н x ( t, t 0 ) .
s u s 1 u 1

D ( t, t 0 ) = н

н

f 0 f 1 dt I [ g ( x, t ) ] .

Suppose that the sign of D(t, t0) alternates, i.e., the separatrices intersect (see Fig. 1d). We seek a perturbation f*(, t) that eliminates the intersection of the separatrices:1 x = f 0 ( x ) + [ f 1 ( x, t ) + f * ( , t ) ] , where f * ( , t ) = ( f * ( , t ), f * ( , t ) ) . 1 2 We denote by [s1 , s2] the interval where the sign of D(t, t0) alternates. Two cases can arise when the system is perturbed by f*(, t): D * ( t, t 0 ) > s or D * ( t, t 0 ) < s 1 , (6)
2

(4)

(5)

The Melnikov distance is the projection of r on the direction normal to the unperturbed separatrix x0 at an instant t.

where D*(t, t0) is the Melnikov distance for system (4). Suppose that (5) is satisfied. (A similar analysis can be
1

We tentatively call f* a regularizing perturbation. Vol. 98 No. 5 2004

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

SUPPRESSION OF CHAOS IN THE VICINITY OF A SEPARATRIX

1047

performed when inequality (6) holds.) Then, I [ g ( x, t ) ] + I [ g * ( , x, t ) ] > s 2 . where
+

Thus, dynamics of systems that can be represented as (1), (3) are regularized by the perturbation (7)
нi t

f * ( , t )

I [ g * ( , x, t ) ] = н

н

e it = Re ------------------------------------ ( I [ g ( x, t ) ] н const ) e d . f 01 ( x ) н f 02 ( x ) f 0 f * dt .
н

By virtue of (7), there exists such that I [ g ( x, t ) ] + I [ g * ( , x, t ) ] = s 2 + = const , , s 2 Hence, I [ g * ( , x, t ) ] = const н I [ g ( x, t ) ] . On the other hand,
+

Next, we explore the possibility of suppressing chaotic dynamics for systems governed by equations of the form x = P ( x, y ) , y = Q ( x, y ) + [ f ( , t ) + F ( x, y ) ] , (10)

.

(8)

where f(, t) is a periodic perturbation; P(x, y), Q(x, y), and F(x, y) are smooth functions; and is a damping parameter. We consider the most common case when a single hyperbolic point is located at the origin (x = y = 0) and P(x, y) = y. Let x0(t) be the solution on the separatrix. For perturbed system (10), the Melnikov distance can be represented as

I [ g * ( , x, t ) ] = н

н

f 0 f * dt .

(9)

D ( t, t 0 ) = н y 0 ( t н t 0 )
н

(11)

Suppose that the function f*(, t) is absolutely integrable over an infinite interval and Fourier transformable. We define f*(, t) as ^ f * ( , t ) = Re { A ( t ) e
нi t

т [ f ( , t ) + F ( x 0, y 0 ) ] dt I [ g ( , ) ] , where y0(t) = x 0 (t). As in the case of Eq. (1), assume that the sign of the Melnikov distance for system (10) alternates, i.e., the separatrices intersect. We seek a perturbation f*(, t) that eliminates chaotic dynamics: x = y, (12) y = Q ( x, y ) + [ f ( , t ) + F ( x, y ) + f * ( , t ) ] . Since system (10) is parameterized by , chaos must be suppressed for each particular value of the parameter. Accordingly, we can write I[g()] instead of I[g(, a)]. For system (12), f
01

}

^ with A ( t ) = (A(t), A(t)), i.e., assume that the regularizing perturbations applied to both components of Eq. (4) are identical. Therefore,

н

н

^ f 0 { A(t )e

нi t

} dt = const н I [ g ( x, t ) ] .

The inverse Fourier transform yields

= y,

f

02

= Q ( x, y ) ,

^ A ( t ) = ( 0, A ( t ) ) .

^ f 0 A(t ) = Hence,

н

( I [ g ( x, t ) ] н const ) e d .

it

Therefore, 1 it A ( t ) = -------------------- ( I [ g ( ) ] н const ) e d . y0 ( t н t0 )

1 A ( t ) = -----------------------------------f 01 ( x ) н f 02 ( x )

н

Thus, a regularizing perturbation for system (12) can be represented as f * ( , t ) e it = Re -------------------- ( I [ g ( ) ] н const ) e d . y0 ( t н t0 )
н нi t

т

н

( I [ g ( x, t ) ] н const ) e d .

it

The quantity A(t) can be interpreted as the amplitude of a regularizing perturbation.
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 98

No. 5

2004

1048

LOSKUTOV, DZHANOEV

Now, let us find a regularizing perturbation in the case when the Melnikov function D(t, t0) admits an additive shift from its critical value. Again, we analyze the case when (5) is satisfied. Suppose that c corresponds to the critical value of the Melnikov function, I c = I [ g ( ,
=
c

In the general case, if f0 = (f01(x), f02(x)), then we obviously obtain 4a(t ) f * ( , t ) = н ------------------------------------ cos ( t ) . f 01 ( x ) н f 02 ( x ) 4. APPLICATION TO PHYSICAL SYSTEMS Now, we use the approach presented above to analyze the DuffingнHolmes-oscillator and pendulum equations. Transverse intersections of stable and unstable manifolds of these unperturbed systems give rise to homoclinic or heteroclinic orbits. 4.1. DuffingнHolmes Oscillator The forced DuffingнHolmes oscillator with a parametrically perturbed cubic term is described by the equation x н x + [ 1 + cos ( t ) ] x = [ cos ( t ) н x ] , (16)
3

)].

Then, a subcritical Melnikov distance can be expressed as I
out

= Ic н ,

where a + is a constant. Assuming that the system perturbed by f *(, t) exhibits regular behavior, we have I' + I where
+ out

+ I [ g * ( ) ] > s2 .

(13)

I [g*()] = н

н

y 0 ( t н t 0 ) f * ( , t ) dt .

On the other hand, it is obvious that we can take any I' a fortiori greater than Ic: I' = I c + a > s 2 . (14)

where and are the amplitude and frequency of the parametric perturbation, respectively, and 1. We rewrite it as (17) 3 v = x н x + [ cos ( t ) н x cos ( t ) н v ] .
3

x = v,

Now, equating the left-hand sides of (13) and (14), we obtain I[g*()] = 2a. Substituting f * ( , t ) = Re { A ( t ) e } , into the expression for I[g*()], we find
it

The corresponding unperturbed Hamiltonian is v x x H 0 = ----- н ---- + ------- . 4 22
2 2 4

нe
н

it

A ( t ) y 0 ( t н t 0 ) dt = 2 a .

The inverse Fourier transform yields

Setting H0 , we find that system (17) has a single hyperbolic point (x = v = 0) with a single separatrix. The solution on the separatrix can be represented as [21] (see also [12н15]) 2 x 0 ( t ) = ------ cosh t , 2 sinh t v 0 ( t ) = x 0 ( t ) = н ------ -------------- . 2 cosh t (18)

A ( t ) y0 ( t н t0 ) = н2 a e
н

нi t

d .

Hence,

(19)

2a A ( t ) = н ----------- e y0 ( t )
н

нi t

4a(t ) d = н -------------------- . y0 ( t н t0 )

Comparing this system with (1), we write f
01

= v, f
02

f

11

= 0,
3

Thus, dynamics of systems that admit additive shift from the critical value of the Melnikov function D(t, t0) are regularized by the perturbation 4a(t ) f * ( , t ) = н -------------------- cos ( t ) , y0 ( t н t0 ) where (t) is the Dirac delta function. (15)

= x н x ,
3

f

12

= cos ( t ) н x cos ( t ) н v .

Therefore, f 0 f 1 = v 0 [ cos ( t ) н x 0 cos ( t ) н v 0 ]
3

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 98

No. 5

2004

SUPPRESSION OF CHAOS IN THE VICINITY OF A SEPARATRIX

1049

and (2) becomes
+

is
+

D ( t, t 0 ) = н dt [ v 0 ( t н t 0 ) cos ( t )
н

(20)

D * ( t, t 0 ) = н To find A(t), we define

н

A(t )v 0(t )e

it

dt .

(23)

н x ( t н t 0 ) v 0 ( t н t 0 ) cos ( t ) н v ( t н t 0 ) ] .
3 0 2 0

Changing to the integration variable = t н t0 , we finally obtain [12н15] sin ( t 0 ) 22 D ( t, t 0 ) = --------- ----------------------------cosh ( /2 ) sin ( t 0 ) 4 4 2 н ------ ( + 4 ) ---------------------------- + ------ . 6 sinh ( /2 ) 3 The sign of D(t, t0) is preserved if 6 d cosh ( /2 ) ---------------------------------------- = 4 2 ( + 4 )
min

D * ( t, t 0 ) + D out ( t, t 0 ) D' ( t, t 0 ) . Since the perturbation f *(, t) is regularizing by assumption, it holds that D' ( t, t 0 ) > D c ( t, t 0 ) . It is obvious that we can take any D'(t, t0) a fortiori greater than Dc(t, t0): 22 D' ( t, t 0 ) = --------- ----------------------------- sin ( t 0 ) cosh ( /2 ) 4 + ------ н d sin ( t 0 ) + a . 3 On the other hand, we can use (23) to write
+

(21)

<

max

(24)

1 6 2 sinh ( /2 ) -4 = ---- --------------------------- ----------------------------- , 2 2 p ( + 4 ) cosh ( /2 )

(22)

where p is an integer (see [12н15]). Using the left-hand inequality in (22), we determine the critical value of the Melnikov function: 22 D c ( t, t 0 ) = --------- ----------------------------- sin ( t 0 ) cosh ( /2 ) 4 + ------ н d sin ( t 0 ) . 3 (An analogous calculation can be performed for the right-hand inequality.) Then, a subcritical value D out ( t, t 0 ) < D c ( t, t 0 ) . can be represented as 22 D out ( t, t 0 ) = --------- ----------------------------- sin ( t 0 ) cosh ( /2 ) 4 + ------ н d sin ( t 0 ) н a , 3 where a > 0 is a constant. Since the perturbation required to regularize the dynamics of system (16) has the form f * ( , t ) = Re { e
+ it

D' ( t, t 0 ) = н

н

A(t )v 0(t )e

нi t

dt (25)

22 4 + --------- ----------------------------- sin ( t 0 ) + ------ н d sin ( t 0 ) н a . cosh ( /2 ) 3 Equating (24) to (25), we have
+

н

A(t )v 0(t )e

it

dt = н 2 a .

The inverse Fourier transform yields
+

2a A ( t ) = н -----------v 0(t )

н

e

нi t

d .

Therefore, dynamics of the forced DuffingнHolmes oscillator are regularized by the perturbation 4a(t ) f * ( , t ) = н ---------------------- cos ( t ) . v 0( t н t0 ) 4.2. Pendulum The analysis presented above can be extended to the classical nonlinear pendulum, whose separatrices make up a heteroclinic orbit in the absence of damping. A periodically forced, damped pendulum is described by the equation [22] x + x + sin x = cos ( t ) .
Vol. 98 No. 5 2004

(26)

A(t )},

the corresponding Melnikov distance D * ( t, t 0 ) = н v 0 ( t ) f * ( , t ) dt .
н

(27)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

1050 . x 0.8 0.6 0.4 0.2 0 н0.2 н0.4 н0.6 н0.8 н0.8 н0.6 н0.4 н0.2 x 0 0.2 0.4

LOSKUTOV, DZHANOEV . x 4 3 2 1 0 н1 н2 0.6 0.8 н3 н4 2 4 6 8 x 10 12 14 16

Fig. 2. Phase portrait of DuffingнHolmes oscillator (16): = 0.145, = 8, = 0.03, = 0.14, = = 1.1.

Fig. 3. Phase portrait of pendulum (27): = 0.04, = 1.35, = 1.0.

. x 3 2 1 0 н1 н2 н3 н1.0 н0.5 0 x
Fig. 4. Phase portrait of DuffingнHolmes oscillator (31): = 0.145, = 8, = 0.03, = 0.14, = = 1.1, a = 2.

. x 3 2 1 0 н1 н2 н3 1.5 1.0 1.5 н35 н34 н33 н32 x
Fig. 5. Phase portrait of pendulum (32): = 0.04, = 1.35, = 1.0, a = 1.2.

н31

н30

н29

н28

The corresponding unperturbed Hamiltonian is x H 0 = ---- н cos x . 2 The phase portrait of the pendulum is 2-periodic in x, with hyperbolic points at (▒, x) and a center at (0, 0). The system has oscillatory, rotatory, and separatrix solutions. We focus here on solutions of the last type: tanh t x 0 ( t ) = ▒ ------------ , cosh t 2 x 0 ( t ) = ▒ ------------ . cosh t
2

The Melnikov distance corresponding to (27) is [22]
+

D ( t 0, ) = н
+

н

( x 0 ( t ) ) dt
2

(28)

▒ cos ( t 0 )

н

sin ( x 0 ( t ) ) x 0 ( t ) cos t dt .

Calculating the integrals, we obtain 1 2 D ( t 0, ) = н 4 B --, 1 ▒ ------------------------ cos ( t 0 ) , (29) 2 cosh -----2
Vol. 98 No. 5 2004

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

SUPPRESSION OF CHAOS IN THE VICINITY OF A SEPARATRIX S 103 (a) 101

1051

10н1

10н3

10н5

10н7 103 (b) 101

10н1

10н3

10н5

10н7 10н1

100

n

101

Fig. 6. Spectral density of a realization x(t) for (a) original DuffingнHolmes oscillator (16) with = 0.145, = 8, = 0.03, = 0.14, and = = 1.1 and (b) regularized DuffingнHolmes oscillator (31) with a = 2.

where B(r, s) is Euler's beta function. Since this Melnikov function D(t0, ) obviously admits additive shift from its critical values, chaotic behavior of the pendulum is suppressed by the perturbation 4a(t ) f * ( , t ) = н -------------------- cos ( t ) , x0 ( t н t0 ) (30)

Physically, the results obtained here mean that dynamics of the DuffingнHolmes oscillator and pendulum are regularized by series of "kicks." 4.3. Numerical Results In the preceding section, it is shown that chaos in DuffingнHolmes-oscillator and pendulum dynamics can be suppressed by applying perturbations (26) and (30), respectively. In this section, we present the results of a numerical analysis. We consider Eqs. (16) and (27). In dynamics of the DuffingнHolmes oscillator, the onset of chaos correVol. 98 No. 5 2004

where x 0 (t) is the solution on the unperturbed sepa ratrix.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

1052 S 105 103 101 10н1 10н3 104 (b) 102

LOSKUTOV, DZHANOEV

both oscillator and pendulum approach regular regimes represented by periodic orbits. To analyze systems (31) and (32) in more detail, we invoke the spectral density defined as 1 2 S ( ) = lim --------- X ( ) , T 2 T where X() is the Fourier transform of a solution x(t) to system (16) or (27). The spectral density provides a simple, but reliable characterization of dynamics of a system under study. It can readily be used to find out whether a motion is regular or chaotic. Figures 6a and 7a show the spectral densities calculated for original systems (16) and (27), respectively; Figs. 6b and 7b, the spectral densities for systems subject to perturbations (26) and (30), respectively. These results demonstrate that chaos is suppressed and dynamics of both systems are regularized. Taking different parameter values corresponding to chaotic behavior, one can find appropriate regularizing perturbations (see above) and obtain qualitatively similar results, i.e., change from chaotic states to regular oscillations. Thus, our numerical analysis is consistent with the analytical results obtained in Section 3. 5. CONCLUSIONS
1 10 wn

100

10н2

10н4

Fig. 7. Spectral density of a realization x(t) for (a) original pendulum equation (27) with = 0.04, = 1.35, and = 1.0 and (b) perturbed equation (32) with a = 1.2.

sponds to the breakdown of a figure-of-eight separatrix. Figure 2 illustrates the structure of a typical chaotic set obtained in this case. The onset of chaos in pendulum dynamics is associated with the breakdown of a heteroclinic trajectory (see Fig. 3). Consider the DuffingнHolmes oscillator and pendulum with additional perturbations (26) and (30), respectively. The corresponding equations are x н x + [ 1 + cos ( t ) ] x = cos ( t ) н x
3

Separatrix splitting is a very convenient method for examining dynamical systems, because it can be used to obtain nonintegrability conditions for many applied problems in analytical form [23]. Currently, the problem of chaos suppression considered in this study is mainly solved by numerical methods (e.g., see [1н10]). However, asymptotic behavior of trajectories can be examined analytically. As a result, the distance between the separatrices split by a perturbation can be found in general form by applying a perturbation method in the vicinity of a homoclinic trajectory. In this study, separatrix splitting is applied to explore the possibility of chaos suppression in dissipative systems. Analytical expressions are obtained for regularizing perturbations. These results are sufficiently general to be applied to various dynamical systems that admit separatrix splitting. REFERENCES
1. S. Boccaletti, C. Grebogi, Y.-C. Lai, et al., Phys. Rep. 329, 103 (2000). 2. A. Loskutov and S. D. Rybalko, in Proceedings of the 5th International Conference on Difference Equations and Applications, ICDEA'2000, Temuco, Chile, 2000 (Taylor and Francis, London, 2002), p. 207. 3. V. V. Alekseev and A. Yu. Loskutov, Vestn. Mosk. Univ., Ser. 3: Fiz. Astron. 26 (3), 40 (1985).
Vol. 98 No. 5 2004

cosh ( t н t 0 ) +2 2 ----------------------------- a ( t ) cos t sinh ( t н t 0 )
2

(31)

x + x + sin x = cos ( t ) +2 cosh ( t н t 0 ) a ( t ) cos ( t ) .

(32)

Figures 4 and 5 show numerical solutions to systems (31) and (32), respectively. It is clear that the dynamics of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

SUPPRESSION OF CHAOS IN THE VICINITY OF A SEPARATRIX 4. V. V. Alekseev and A. Yu. Loskutov, Dokl. Akad. Nauk SSSR 293, 1346 (1987) [Sov. Phys. Dokl. 32, 270 (1987)]. 5. E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990). 6. F. J. Romeiras, E. Ott, C. Grebogi, and W. P. Dayawansa, Physica D (Amsterdam) 58, 165 (1992). 7. T. Shinbrot, C. Grebogi, E. Ott, and J. A. Yorke, Nature 363, 411 (1993). 8. T. Shinbrot, Adv. Phys. 44, 73 (1995). 9. E. Ott and M. L. Spano, Phys. Today 48, 34 (1995). 10. A. Loskutov, Comput. Math. Mod. 12, 314 (2001). 11. V. K. Mel'nikov, Tr. Mosk. Mat. Oнva 12, 3 (1963). 12. R. Lima and M. Pettini, Phys. Rev. A 41, 726 (1990). 13. F. Cuadros and R. Chac╤n, Phys. Rev. E 47, 4628 (1993). 14. R. Lima and M. Pettini, Phys. Rev. E 47, 4630 (1993). 15. R. Chac╤n, Phys. Rev. E 51, 761 (1995).

1053

16. Y. Kivshar, F. Rodelsperger, and H. Benner, Phys. Rev. E 49, 319 (1994). 17. R. Chac╤n, Phys. Rev. Lett. 86, 1737 (2001). 18. G. Cicogna and L. Fronzoni, Phys. Rev. A 42, 1901 (1990). 19. V. G. Gel'freoekh and V. F. Lazutkin, Usp. Mat. Nauk 56, 79 (2001). 20. P. J. Holmes and J. E. Marsden, Commun. Math. Phys. 82, 523 (1982). 21. J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer, Berlin, 1990). 22. J. L. Trueba, J. Rams, and M. A. F. Sanju└n, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2257 (2000). 23. V. V. Kozlov, Symmetry, Topology, and Resonances in the Hamilton Mechanics (Udmurt. Gos. Univ., Izhevsk, 1995).

Translated by A. Betev

Spell: ok

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 98

No. 5

2004