Analytical description of the structure of chaos

Abstract

We consider analytical formulae that describe the chaotic regions around the main periodic orbit $(x=y=0)$ of the Hénon map. Following our previous paper (Efthymiopoulos, Contopoulos, Katsanikas $2014$) we introduce new variables $(ξ, η)$ in which the product $ξη=c$ (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation $Φ$ to the plane $(x,y)$, giving "Moser invariant curves". We find that the series $Φ$ are convergent up to a maximum value of $c=c_{max}$. We give estimates of the errors due to the finite truncation of the series and discuss how these errors affect the applicability of analytical computations. For values of the basic parameter $κ$ of the Hénon map smaller than a critical value, there is an island of stability, around a stable periodic orbit $S$, containing KAM invariant curves. The Moser curves for $c \leq 0.32$ are completely outside the last KAM curve around $S$, the curves with $0.32<c<0.41$ intersect the last KAM curve and the curves with $0.41\leq c< c_{max} \simeq 0.49$ are completely inside the last KAM curve. All orbits in the chaotic region around the periodic orbit $(x=y=0)$, although they seem random, belong to Moser invariant curves, which, therefore define a "structure of chaos". Orbits starting close and outside the last KAM curve remain close to it for a stickiness time that is estimated analytically using the series $Φ$. We finally calculate the periodic orbits that accumulate close to the homoclinic points, i.e. the points of intersection of the asymptotic curves from $x=y=0$, exploiting a method based on the self-intersections of the invariant Moser curves. We find that all the computed periodic orbits are generated from the stable orbit $S$ for smaller values of the Hénon parameter $κ$, i.e. they are all regular periodic orbits.

22 pages, 9 figures