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Scaling law and asymptotic distribution of Lyapunov exponents in conservative dynamical systems with many degrees of freedom. (English) Zbl 0609.70026
We study by numerical means the infinite product of 2N\(\times 2N\) conservative random matrices which mimics the chaotic behaviour of Hamiltonian systems with \(N+1\) degrees of freedom made of weakly nearest- neighbour coupled oscillators. The maximum Lyapunov exponent \(\lambda_ 1\) exhibits a power-law behaviour as a function of the coupling constant \(\epsilon\) : \(\lambda_ 1\sim \epsilon^{\beta}\) with either \(\beta =\) or \(\beta =2/3\), depending on the probability distribution of the matrix elements. These power laws do not depend on N and moreover increasing N, \(\lambda_ 1\) rapidly tends to an asymptotic value \(\lambda^*_ 1\) which only depends on \(\epsilon\) and on the kind of probability distribution chosen for building up the matrices. We also compute the spectrum of the Lyapunov exponents and show that it has a thermodynamic limit of large N.
This suggests the existence of a Kolmogorov entropy per degree of freedom proportional to \(\lambda^*_ 1\).

MSC:
70K50 Bifurcations and instability for nonlinear problems in mechanics
70-08 Computational methods for problems pertaining to mechanics of particles and systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
82B05 Classical equilibrium statistical mechanics (general)
28D20 Entropy and other invariants
37A99 Ergodic theory
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