Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 H\'{e}non map. Following our previous
paper (Efthymiopoulos, Contopoulos, Katsanikas $2014$) 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 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 $c
\leq 0.32$ 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
$\Phi$. 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 $\kappa$, i.e. they are all regular periodic orbits.

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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an