Mathematics – Dynamical Systems
Scientific paper
Sep 2008
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008nnpr.book..393c&link_type=abstract
New Nonlinear Phenomena Research, Edited by Tomas B. Perlidze, Nova Publishers, p. 393-410
Mathematics
Dynamical Systems
Chaos, Diffusion
Scientific paper
In the present chapter we review as well as provide new results on the processes that lead to chaotic diffusion 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 diffusion in phase space, are the overlap of resonances, resonance crossings and Arnold diffusion-like processes. When dealing with nearly integrable Hamiltonian systems, chaos actually means the variation 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 either its routes or its extent. Though one could get accurate values of the Lyapunov exponents, the KS entropy or any other indicator of the stability of the motion, they only provide local values for the variation of the integrals. A given orbit in a chaotic component of phase space could have, for instance, a positive and large value for two of 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 significant is the extent of the domain and the time scale over which diffusion may occur. In [1] 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 in physical time scales. All these issues as well as a relatively new fast indicator of the dynamics, the Mean Exponential Growth Factor of Nearby Orbits (MEGNO), are thoroughly discussed in this chapter by both numerical and theoretical means.
Cincotta Pablo M.
Giordano Claudia M.
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