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.