Dynamical entropy of the separatrix map

Abstract

We calculate the maximum Lyapunov exponent of the motion in the separatrix map's chaotic layer, along with calculation of its width, as functions of the adiabaticity parameter $\lambda$. The separatrix map is set in natural variables; and the case of the layer's least perturbed border is considered, i.~e., the winding number of the layer's border (the last invariant curve) is the golden mean. Although these two dependences (for the Lyapunov exponent and the layer width) are strongly non-monotonous and evade any simple analytical description, the calculated dynamical entropy $h$ turns out to be a close-to-linear function of $\lambda$. In other words, if normalized by $\lambda$, the entropy is a quasi-constant. We discuss whether the function $h(\lambda)$ can be in fact exactly linear, $h \propto \lambda$. The function $h(\lambda)$ forms a basis for calculating the dynamical entropy for any perturbed nonlinear resonance in the first fundamental model, as soon as the corresponding Melnikov--Arnold integral is estimated.