Analytical description of the structure of chaos

We consider analytical formulae that describe the chaotic regions around the main periodic orbit (x = y = 0) of the Henon map. Following our previous paper (Efthymiopoulos et al 2014 Celest. Mech. Dyn. Astron. 119 331) we introduce new variables (xi, eta) in which the product xi eta = c (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation Phi to the plane (x, y), giving 'Moser invariant curves'. We find that the series Phi 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 kappa of the Henon 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 <= 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 <= c < c(max) 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 F. 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 Henon parameter., i.e. they are all regular periodic orbits.