Topics on diffusion in phase space of multidimensional Hamiltonian systems (capítulo invitado)

PABLO CINCOTTA; CLAUDIA GIORDANO

New Nonlinear Phenomena Research

Nova Science Publishers

Lugar: New York; Año: 2008; p. 319 - 337

In the present chapter we review as well as provide new results on the processesthat lead to chaotic diffusion in phase space of multidimensionalHamiltonian systems.It is well known that the simplest mechanisms leading to a transition from regularityto chaos, and therefore to diffusion in phase space, are the overlap of resonances,resonance crossings and Arnol’d diffusion–like processes.When dealing with nearly integrable Hamiltonian systems, chaos actually meansthe variation of the unperturbed integrals, which is usually called chaotic diffusion.Unfortunately, it does not yet exist any theory that could describe global diffusion inphase 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 orany other indicator of the stability of the motion, they only provide local values forthe variation of the integrals. A given orbit in a chaotic component of phase spacecould 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 changeover a rather large domain. This is a natural consequence of the structure of phasespace 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 thosesuitable for the description of an elliptical galaxy, the time–scale over which diffusionbecomes relevant is several orders of magnitude the Hubble time. On the other hand,in models corresponding to planetary or asteroidal dynamics, diffusion may occur inphysical time–scales.All these issues as well as a relatively new fast indicator of the dynamics, theMeanExponential Growth Factor of Nearby Orbits (MEGNO), are thoroughly discussed inthis chapter by both numerical and theoretical means.NOTA: EL FULL TEXT HA SIDO LIGERAMENTE MODIFICADO, SIN AFECTAR EL CONTENIDO CIENTÍFICO DEL MISMO Y PARA QUE OCUPE MENOS DE 20 MB. SE HAN SUPERPUESTO DOS FIGURAS Y EN CONSECUENCIA EL ARTÍCULO INCLUIDO EN ESTA BASE DE DATOS CUENTA CON UNA PÁGINA MENOS QUE EN LA PUBLICACIÓN ORIGNAL.