We consider analytical formulae that describe the chaotic regions around the
main periodic orbit (x=y=0) of the H\'{e}non 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=cmax​. 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 H\'{e}non 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<cmax​≃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 H\'{e}non
parameter κ, i.e. they are all regular periodic orbits.Comment: 22 pages, 9 figure