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International Journal of Bifurcation and Chaos, Vol. 17, No. 10 (2007) 36033606 c World Scientific Publishing Company

STABILIZATION OF CHAOTIC BEHAVIOR IN THE RESTRICTED THREE-BODY PROBLEM
ARSEN DZHANOEV and ALEXANDER LOSKUTOV Physics Department, Moscow State University, Leninskie Gory, Moscow 119992, Russia janoev@pol ly.phys.msu.ru Loskutov@chaos.phys.msu.ru Received Octob er 19, 2005; Revised February 20, 2006
The restricted three-bo dy problem on the example of a perturbed Sitnikov case is considered. On the basis of the Melnikov metho d we study a possibility to stabilize the obtained chaotic solutions by two bo dies placed in the triangular Lagrange points. It is shown that in this case, in addition to regular and chaotic solutions, there exist stabilized solutions. Keywords : Restricted three-body problem; Sitnikov problem; Melnikov metho d.

1. Introduction
Henri Poincarґ in his work on Celestial Mechanics e underlined a p ossibility of chaotic b ehavior in threeb ody problem by the destruction of homoclinic contours. Later, an existence of transverse homoclinic p oints in the three-b ody problem was analytically verified. A well-known modification of the restricted three-b ody problem is that of Sitnikov [1960]. The Sitnikov problem takes place when two equal masses M orbit around their barycentre. A third, massless or small but finite mass b ody (particle) µ moves in their gravitational field p erp endicular to the motion surface of the primaries (see Fig. 1). It can b e shown, that the oscillation of the third b ody is chaotic (under certain additional conditions). In this work we consider the problem of stabilization of this chaotic b ehavior. In general, this problem is related to stabilization and control of unstable and chaotic b ehavior of dynamical systems by external forces. A comprehensive study of chaotic systems with external controls may thus provide a key to the understanding many nonlinear processes in b oth localized and distributed systems. This could b e of interest, for example, in the study of planetary systems of binary stars.
µ

z
z'

Fig. 1.

The Sitnikov problem.

2. Sitnikov Problem
With the prop er time and space scaling we can write the corresp onding differential equation for the particle µ in the form z=- Ё z , (z + 2 )3/2
2

(1)

where = (t) = 1 + cos t + O(2 ). Here the small parameter is closely related to the eccentricity. To

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A. Dzhanoev & A. Loskutov

make clear how this problem is related to homoclinical orbits, we introduce noncanonical transformation [Dankovicz & Holmes, 1995]: z = tan u, v = z, u [-/2, /2], v R. Then the Hamiltonian for Eq. (1) in the new variables (u, v ) has the form: 1 1 H (u, v ) = v 2 - 2 u + 2 2 tan = H0 (u, v )+ H1 (u, v , t, ), (2)
Fig. 2.

µ

m

L4

L5

where H0 (u, v ) = 1/2v 2 - cos u. As we can see, when = 0 our system reveals the dynamics of a nonlinear p endulum, which oscillates with increasing amplitude in time. If = 0 then the system (1) exhibits chaotic prop erties [Dankovicz & Holmes, 1995].

m

Stabilization of particle's chaotic b ehavior.

3. Stabilization of Chaotic Behavior
As follows from the work of V. M. Alekseev [2001] one can find one-to-one corresp ondence b etween the collinear solutions set of the Sitnikov problem and the symb olic set : v - in1 | ... mn-1 in-1 mn in mn+1 in+1 ... |v + , where mn1 , n1 < n < n2 are natural numb ers N , in = 0 or 1, v ± [0,], n1 < 0 n2 . The numb er of collinear configurations on the solution set is defined by in . Symb ol mn indicates the numb er of total rotations of the mass M bodies between (n - 1)-th and nth system collinear configuration. Therefore, we may choose mn such that our system oscillates slowly nearby its origin (when = 0). At = 0 (a p erturb ed chaotic nonlinear p endulum) there are transverse homoclinic p oints. To clarify the sense of this statement, let us consider a symb olic sequence [Alekseev, 2001]: - 0N1 1N1 0 ... 1N1 0N2 0 ... N2 0N1 1N1 0N1 1
2k1 times 2k2 times +

binary star. Finally, having returned to the initial orbit, the spacecraft µ made one and half oscillation near this orbit. Then it moved away to infinity with velocity + following the same direction from which it came initially. Now, placing two new b odies of the mass m (here M m µ) in the Lagrange p oints L4 and L5 we can achieve the situation when the influence of these b odies on the mass µ has a form of p eriodical (forced) impulses (the tra jectories of b odies of mass m are shown in Fig. 2). Therefore, this may b e treated as a nonlinear p erturb ed p endulum with a sp ecific external force. Earlier it has b een shown [Loskutov & Dzhanoev, 2004] that using Melnikov method, we can obtain the stabilized dynamics for µ. In this case, duration time of these impulses should b e much less than the characteristic time T of the system, i.e. T and 0. Thus, the stabilized system has the following form: H (u, v ) = H0 (u, v ) + H1 (u, v , t)+
n

(t - n ) .

(3)

This sequence can b e interpreted in the following manner. A spacecraft µ which arrived at a binary star system from infinity with velocity - , first came to the nearly p eriodic orbit with p eriod 4N1 . In this orbit, it made k1 complete oscillations. During each of these oscillations, the spacecraft returned twice to the mass centre of the binary star in the moments of maximal and minimal distances b etween the comp onents of the binary star. Then the spacecraft moved to another nearly p eriodic orbit with p eriod 4N2 . Here it made k2 complete oscillations. Each of the 2k2 spacecraft returns to the mass centre takes place in moments of the minimal distance b etween the comp onents of the

Between pulses the motion of the particle µ is free. Then, sp eaking in terms of celestial mechanics, our system undergoes the influence of some exterior celestial b odies (say, two spacecrafts) which orbit near the third b ody (particle).

4. Numerical Results
The onset of chaos in dynamics of the mass µ in three-b ody system (Fig. 1) corresp onds to the breakdown of a heteroclinic tra jectory. Figure 3 illustrates the structure of a typical chaotic set obtained in this case. In Fig. 4, numerical solutions of the system with Hamiltonian (3) is shown. It is clear that the dynamics of the particle µ tends to a regular regime represented by a p eriodic orbit.

Stabilization of Chaotic Behavior in the Restricted Three-Body Problem

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Fig. 3.

Numerical solution to system with Hamiltonian (2).

Fig. 4.

Numerical solution to system with Hamiltonian (3).

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A. Dzhanoev & A. Loskutov

The analysis of invariant characteristics of chaoticity (Lyapunov exp onents, p ower sp ectra, etc.) confirms this conclusion.

the triangular Lagrange p oints. It is shown that in this case, in addition to regular and chaotic solutions, there exist zones of the stabilized b ehavior.

5. Concluding Remarks
Separatrix splitting is a very convenient method for examining dynamical systems b ehavior, b ecause it can b e used to obtain nonintegrability conditions for many applied problems in an analytical form. As a result, the distance b etween the splitting separatrices can b e found by applying a p erturbation method in the vicinity of a homoclinic tra jectory. In this study, separatrix splitting is applied to explore the p ossibility of chaos suppression (stabilization) [Alekseev & Loskutov, 1987; Ott et al., 1990] in the p erturb ed restricted three-b ody problem (so-called Sintikov case). On the basis of the Melnikov method, it is found that stabilization of chaotic solutions can b e obtained by placing two b odies in

References
Alekseev, V. V. & Loskutov, A. [1987] "Control of the system with strange attractor by perio dical parametric perturbation," Sov. Phys. Dok. 32, 270273. Alexeev, V. M. [2001] Lectures on Celestial Mechanics (Moscow-Izhevsk). Dankovicz, H. & Holmes, Ph. [1995] "The existence of transverse homo clinic points in the Sitnikov problem," J. Diff. Eqs. 116, 468483. Loskutov, A. & Dzhano ev, A. [2004] "Suppression of chaos in the vicinity of a separatrix," J. Exper. Th. Phys. (JETP ) 125, 191200. Ott, E., Grebogi, C. & Yorke, J. A. [1990] "Controlling chaos," Phys. Rev. Lett. 64, 11961199. Sitnikov, K. [1960] "Existence of oscillating motions in three-bo dy problem," Sov. Phys. Dokl. 133, 303306.