Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles

Abstract

This paper studies the behavior under iteration of the maps T_(ab) (x, y) = (F_(ab) (x) − y, x) of the plane ℝ^2, in which F_(ab) (x) = ax if x ≥ 0 and bx if x < 0. The orbits under iteration correspond to solutions of the difference equation x_(n+2) = ½(a-b)|x_(n+1)| + ½(a+b)x_(n+1) – x_n. This family of piecewise-linear maps of the plane has the parameter space (a,b) ϵ ℝ^2. These maps are area-preserving homeomorphisms of ℝ^2 that map rays from the origin into rays from the origin. We show the existence of special parameter values where T_(ab) has every nonzero orbit contained in an invariant circle with an irrational rotation number, with invariant circles that are piecewise unions of arcs of conic sections. Numerical experiments indicate the possible existence of invariant circles for many other parameter values.