Instability of an area-preserving polynomial mapping

Abstract

Fixed points and eigencurves have been studied for the Hénon-Heiles mapping:x′=x+a (y−y ³),y′=y(x′−x′ ³). Eigencurves of order 21 proceed rapidly to infinity fora=1.78, but as ‘a’ decreases, they spiral around the origin repeatedly before escaping to infinity. Fixed pointsx _f on thex-axis have been located for the range 1≤a≤2.4, for ordersn up to 100. Their locations vary continuously witha, as do the eigencurves, and hyperbolic points remain hyperbolic.

Forn=3 and 2.4≥a≥2.37, a very detailed study has been made of how escape occurs, with segments of an eigencurve mapping to infinity through various escape channels. Further calculations with ‘a’ decreasing to 2.275 show that this instability is preserved and that the eigencurve will spiral many times around the origin before reaching an escape channel, there being more than 34 turns fora=2.28. The rapid increase of this number is associated with the rapid decrease of the intersection angle between forward and backward eigencurves (at the middle homoclinic point), with decreasing ‘a’, this angle governing the outward motion.

By a semi-topological argument, it is shown that escape must occur if the above intersection angle is nonzero. In the absence of a theoretical expression for this angle, one is forced to rely on the numerical evidence. If the angle should attain zero for a valuea=a _c>a_m,wherea _m .is the minimum value for which the fixed points exist, then no escape would be possible fora<a _c However, on the basis of calculations by Jenkins and Bartlett (1972) forn=6, and the results of the present article forn=3, it appears highly probable thata _c=a_m,and that escape from the neighborhood of a hyperbolic point is always possible.

If there is escape from the hyperbolic fixed point forn=4,a=1.6, located atx _f=0.268, then the eigencurve must cross the apparently closed invariant curve of Hénon-Heiles which intersects thex-axis atx≊±0.4, so that this curve cannot in fact be closed.