Topics on diffusion in phase space of multidimensional Hamiltonian systems

CINCOTTA, P.M.; GIORDANO, C.M.

New Nonlinear Phenomena Research

NOVA SCIENCE PUBLISHERS, INC.

Lugar: Hauppauge, NY 11788, novaeditorial@earthlink.net; Año: 2007; p. 319 - 337

TOPICS ON DIFUSSION IN PHASE SPACE OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS P. M. Cincotta and C. M. Giordano. Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata (CONICET), Paseo del Bosque, 1900 La Plata, Argentina. ABSTRACT In the present effort we review as well as provide new results on the processes that lead to difussion in phase space of multidimensional Hamiltonian systems. It is well known that the simplest mechanisms leading to a transition from regularity to chaos, and therefore to difussion in phase space, are the overlap of resonances, resonance crossings, and Arnol´d diffusion--like processes (see for instance, Arnold´d 1964; Chirikov 1979). When dealing with nearly integrable Hamiltonian systems, chaos actually means the variations of the unperturbed integrals, which is usually called chaotic diffusion. Unfortunately it does not yet exist any theory that could describe global diffusion in phase space. In other words, it is not possible to estimate neither its rate nor its direction or route. Though one could get accurate values of the Lyapunov exponents, the KS entropy, the MEGNO (Cincotta et al. 2003; Cincotta and Sim´o 2000) or any other indicator of the stability of the motion, they only provide local values of the diffusion rate. A given orbit in a chaotic component of the phase space coud have, for instance, two positive and large values for the Lyapunov exponents, however this does not necessarily mean that the unperturbed integrals would change over a rather large domain. This is a natural consequence of the structure of  phase space of almost all actual dynamical systems such as planetary systems or galaxies. Therefore, what is actually relevant is the extent of the domain and the time--scale over which  diffusion may occur. In Giordano and Cincotta 2004, it is shown that in models similar to those suitable for the description of an elliptical galaxy, the time--scale over which diffusion becomes relevant is several orders of magnitude the Hubble time. On the other hand, in models corresponding to planetary or asteroidal dynamics, diffusion may occur over physical time--scales. All these issues are thoroughly discussed in the present chapter by both numerical and theoretical means. REFERENCES Arnol´d, V.I., 1964, Sov. Math.-Dokl., 5, 581. Chirikov, B., 1979, Phys. Rep. 52, 63. Cincotta, P.M., 2002, New AR 46, 13. Cincotta, P.M. and Simo, C., 2000, A&AS, 147, 205. Cincotta, P.M., Giordano, C.M. and  Simo C., 2003, Phys. D, 182, 151. Giordano, C.M. and Cincotta, P.M., 2004, A&A, 423, 745.