Lyapunov exponents in resonance multiplets

• The motion in resonance multiplets in Hamiltonian dynamics is considered.
• Analytical approaches for estimating the maximum Lyapunov exponent (MLE) are developed.
• The MLE in the uniform doublet is shown to provide a lower bound for MLEs in multiplets.
• The MLE in the uniform infinitet is shown to provide an upper bound.

The problem of estimating the maximum Lyapunov exponents of the motion in a multiplet of interacting nonlinear resonances is considered for the case when the resonances have comparable strength. The corresponding theoretical approaches are considered for the multiplets of two, three, and infinitely many resonances (i.e., doublets, triplets, and “infinitets”). The analysis is based on the theory of separatrix and standard maps. A “multiplet separatrix map” is introduced, valid for description of the motion in the resonance multiplet under certain conditions. In numerical experiments it is shown that, at any given value of the adiabaticity parameter (which controls the degree of interaction/overlap of resonances in the multiplet), the value of the maximum Lyapunov exponent in the multiplet of equally-spaced equally-sized resonances is minimal in the doublet case and maximal in the infinitet case. This is consistent with the developed theory.