Документ взят из кэша поисковой машины. Адрес оригинального документа : http://chaos.phys.msu.ru/loskutov/PDF/Springer_Dzh_How_San_Losk.pdf Дата изменения: Mon Jan 31 16:11:41 2011 Дата индексирования: Mon Oct 1 20:17:52 2012 Кодировка:

Stabilized Chaos in the Sitnikov Problem
A.R. Dzhanoev1 , A. Loskutov2 , J.E. Howard3 , and M.A.F. Sґ 1 anju
1

2 3

Departamento de Fisica, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain arsen.dzhanoev@urjc.es Moscow State University, 119992 Moscow, Russia Laboratory for Atmospheric and Space Physics and Center for Integrated Plasma Studies, University of Colorado, Boulder, CO 80309, USA

1 Formulation of the Problem
The Sitnikov problem consists of two equal masses M (called primaries) moving in circular or elliptic orbits about their common center of mass and a third, test mass µ moving along the straight line passing through the center of mass normal to the orbital plane of the primaries. This problem has attracted the attention of many other authors (see for instance [1­9]). The equation of motion can be written in scaled coordinates and time as z+ Ё [(t )2 z + z 2 ]3
/2

= 0,

(1)

where z denotes the position of the particle µ along the z-axis and (t) = 1 + e cos(t) + O(e2 ) is the distance of one primary body from the center of mass. Here we see that the system (1) depends only on the eccentricity, e, which we shall assume to be small. The linear approach to this system with assumption that (1) possesses moderate eccentricity and small amplitudes was carried out in [7]. We first consider the circular Sitnikov problem i.e. when 1 1 ,p = z . The level curves H = h, where e = 0, for which H = p2 - 2 1+ z 2 h [-2, +), partition the phase space (p, z ) into qualitatively different types of orbits. We are interested in solutions that correspond to the level curves H = 0, namely two parabolic orbits that separate elliptic and hyperbolic orbits and can be considered as a separatrix between these two classes of behavior. To make clear how this problem is related to homoclinic orbits, let us employ the non-canonical transformation [9] z = tan u, p = z, u - , ,v R. 22 Then the Hamiltonian for (1) in the new variables (u,p) has the form H (u, p) = 12 1 1 = H0 (u, p)+ eH1 (u, p, t, e), where H0 (u, p) = p2 - cos u. p- 2 + tan2 u 2 2 (t) One can see that when e = 0 the form of the Hamiltonian that obtained after
G. Contopoulos, P.A. Patsis, Chaos in Astronomy. Astrophysics and Space Science Proceedings, doi: 10.1007/978-3-540-75826-6, c Springer-Verlag Berlin Heidelberg 2009

301


302

A.R. Dzhanoev et al.

non-canonical transformation exhibits the pendulum character of motion. In this work we consider only small values of e. Hence due to the KAM-theorem, since our system has 3/2 degrees of freedom the invariant tori bound the phase space and chaotic motion is finite and takes place in a small vicinity of a separatrix layer. Our analysis is directed to the stabilization of this chaotic behavior in the elliptic Sitnikov problem.

2 Stabilization of Chaotic Behavior in the Extended Sitnikov Problem
The idea that chaos may be suppressed goes back to the publications. [10­ 14] We consider the problem of stabilization of chaotic behavior in sys tems with separatrix contours that can be described by (2.1) x = f0 (x) + f1 (x,t), where f0 (x) = (f01 (x),f02 (x)), f1 (x,t) = (f11 (x,t),f21 (x,t)). For this equation the Melnikov distance, which "measures" (in the first order of ) the distance between stable and unstable manifolds (Fig. 1),


D(t0 ) is given by D(t0 ) = -
-

f0 f1 dt I [g (t0 )]. We assume that D(t0 )

changes its sign. tion f (, t) that (2.2) x = f0 (x)+ Suppose D(t0 ) bation f (, t) is where D (t0 ) ­ M (analysis for the


To suppress chaos we should get a function of stabilizaleads to a situation when separatrices are not intersected: [f1 (x,t)+ f (, t)] , where f (, t) = (f1 (, t),f2 (, t)). [s1 ,s2 ] and s1 < 0 < s2 . After the stabilizing perturapplied we have two cases: D (t0 ) > s2 or D (t0 ) < s1 , elnikov distance for system (2.2). We consider the first case second one is similar). Then I [g (t0 )] + I [g (, t0 )] > s2 ,
+

where I [g (, t0 )] = -
-

f0 f dt. This expression is true for all left hand

a)

b)

c)

d)

Fig. 1. Poincarґ section t = const (mod T ) of the system (2.1) for = 0 (a) and e = 0 (b­d). Only in case of (d) we have homoclinical chaos


Stabilized Chaos in the Sitnikov Problem

303

side values of inequality that is greater than s2 . It is derived that I [g (t0 )] + I [g (, t0 )] = s2 + = const, where , s2 IR+ . Therefore I [g (, t0 )] =


const - I [g (t0 )]. On the other hand, I [g (, t0 ))] = -
-



f0 f dt. We choose

f (, t) from the class of functions that are absolutely integrable on an infinite interval such that they can be represented in Fourier integral form. Then ^ ^ f (, t) = Re{A(t)e-it }. Here we suppose that A(t) = (A(t),A(t)) i.e. the regularizing perturbations applied to both components of (2.2) are identical.


Therefore -
-

^ f0 A(t)e-

i t

dt = const - I [g (t0 )]. The inverse Fourier (I [g (t0 )] - const) eit d . Hence,

^ transform yields: f0 A(t) = 1 f01 (x) - f02 (x)
-

A(t) =

(I [g (t0 )] - const) eit d . Here A(t) can be inter-

preted as the amplitude of the "stabilizing" perturbation. Thus, for system (2.1) the external stabilizing perturbation has the form: e-it (I [g (t0 )] - const) eit d . In conservaf (, t) = Re f01 (x) - f02 (x)
-

tive case const=0. From the physical point of view the dynamics of the chaotic system are stabilized by a series of "kicks". The orbit that was chaotic and became regular under influence of the external perturbation we call the stabilized orbit. Let us now consider two bodies of mass m that are placed in the vicinity of the stable triangular Lagrange points of the Sitnikov problem (as shown in Fig. 2). Here we treat only the hierarchical case : µ m M . In the new
Z

Z

m

L

4

M

M

L

5

m

Fig. 2. Geometry of the Extended Sitnikov problem


304

A.R. Dzhanoev et al.

configuration we can achieve the situation when the influence of bodies placed close to the triangular Lagrange points to the particle µ can be presented as a series of periodic impulses. Taking into account the new configuration (Fig. 2) we may say that there is a connection between the extended elliptical Sitnikov problem and the motion of the chaotic nonlinear pendulum with an external impulse-like perturbation. The Hamiltonian of such system changes to
+

H (u, p) = H0 (u, p)+ e H1 (u, p, t, e)+
n=-

(t - n ) ,

(2)

where is the duration of the impulsive forces that the particle µ experiences from bodies in the vicinity of L4 and L5 . Now taking into account the result from the first part of this section we conclude that the forces which the particle experiences from bodies in the neighborhood of L4 and L5 act on the chaotic behavior of µ as an external stabilizing perturbation and the system (2) represents the system with stabilized chaotic behavior that corresponds to the stabilized orbits in the extended Sitnikov problem. The extension of the analysis carried out above to the corrections of higher order in of the (1) and numerical verification of the obtained results could be found in [16]. In summary, on the basis of the elliptic Sitnikov problem we constructed a configuration of five bodies which we called the extended Sitnikov problem and analytically showed that in this configuration along with chaotic and regular orbits a new type of orbit (stabilized) could be realized. We thank Carles Simґ o and David Farrelly for valuable discussions. A. Dzhanoev acknowledges that this work is supported by the Spanish Ministry of Education and Science under the pro ject number SB2005-0049. Financial support from pro ject number FIS2006-08525 (MEC-Spain) is also acknowledged. This work was supported in part by the Cassini pro ject.

References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. W.D. MacMillan: Astron. J. 27, 11 (1913) K. Stumpff: Himmelsmechanik Band II VEB, (Berlin, 1965) pp 73­79 K.A. Sitnikov: Dokl. Akad. Nauk USSR 133, 2, 303 (1960) V.M. Alexeev: Math. USSR Sbornik 5, 73; 6, 505; 7, 1 (1969) J. Moser: Stable and random motions in dynamical systems, (Prinston University Press, Prinston, N.J., 1973) L. Llibre and C. Simґ Publicaciones Matem` o: atiques U.A.B. 18, 49 (1980) J. Hagel and C. Lhotka: Celest. Mech. Dyn. Astron. 93, 201 (2005) R. Dvorak: Celest. Mech. Dyn. Astron. 56, 71 (1993) H. Dankowicz and Ph. Holmes: J. Differ. Equ. 116, 468 (1995) V.V. Alexeev and A. Loskutov: Sov. Phys.-Dokl. 32, 270 (1987) A. Loskutov and A.I. Shishmarev: Chaos 4, 351 (1994) E. Ott, C. Grebogi, and J.A. Yorke: Phys. Rev. Lett. 64, 1196 (1990)


Stabilized Chaos in the Sitnikov Problem

305

13. A. Loskutov: Discr. 14. T. Schwalger, A.R. (2006) 15. A.R. Dzhanoev, A. 7, 275 (2007) 16. A.R. Dzhanoev, A.

Continuous Dyn. Syst., Ser. B 6, 1157 (2006) Dzhanoev, and A. Loskutov: Chaos 16, 2, 023109 Loskutov et al: Discr. Continuous Dyn. Syst., Ser. B Loskutov et al: submitted to Phys. Rev. E, (2008)