第８３回「非線形・統計力学とその周辺」セミナーのご案内

日時：平成１９年４月１３日（金）１４：４５から

場所：京都大学工学部総合校舎１０２講義室（吉田キャンパス）

講演者：T. Bountis
(Department of Mathematics and Director of Center
for Research and Applications of Nonlinear Systems,
University of Patras, Greece)

講演題目：

Geometrical Properties of Local Dynamics in Hamiltonian systems:
The Generalized Alignment Index (GALI) method

講演要旨：

We investigate the dynamics of multidimensional Hamiltonian systems and symplectic maps, by studying volume elements formed by unit deviation vectors about their orbits. The different time evolution of these volumes is used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities called the Generalized Alignment Index of order k (GALIk). We show analytically and verify numerically on particular examples of N degree of freedom Hamiltonian systems that, for chaotic orbits, GALIk tends exponentially to zero with exponents that depend on several Lyapunov exponents. In the case of regular orbits, GALIk is nearly constant for N >= k>=2 and goes to zero for 2N >= k>= N following power laws that depend on the dimension of the torus . We are thus able to: (i) detect chaotic oscillations of N particle systems much faster than other methods, (ii) identify low--dimensional tori of Fermi--Pasta --Ulam lattices at low energies, (iii) identify 2- dimensional tori of quasiperiodic breathers in a lattice without linear dispersion and (iv) predict weak diffusion away from quasiperiodic motion in these lattices, long before it is actually observed in the oscillations.