A physical interpretation of the Hénon map

Abstract

The orientation-preserving Hénon map is shown to be equivalent to the period one return map of a periodically kicked harmonic oscillator with a suitable nonlinear coupling to the kicking term. The conservative, dissipative, and extremely dissipative (logistic limit) cases are each considered. All can be derived from a single differential equation describing the kicked oscillator system. In each case explicit coordinate and parameter transformations are given relating the return map of the kicked harmonic oscillator to a standard form of the Hénon map. In the dissipative case two studies are presented with periodic kicking replaced by a smooth periodic function. Each is found to yield a “Hénon-like” chaotic attractor.