Dynamics of a Family of Piecewise-Linear Area-Preserving Plane Maps I. Invariant Circles

This paper studies the behavior under iteration of the maps Tab(x, y) = (Fab(x)−y, x) of the plane R2, in which Fab(x) = ax if x ≥ 0 and bx if x < 0. This family of piecewise-linear maps has the parameter space (a, b) ∈ R2. These maps are area-preserving homeomorphisms of R2 that map rays from the origin into rays from the origin. The orbits of the map are solutions of the nonlinear difference operator of Schrödinger type −xn+2+2xn+1−xn−Vμ(xn+1)xn+1 = Exn+1, with energy parameter E = 2 − 1 2(a + b) and with an antisymmetric step-function potential Vμ(xn) specified by μ = 1 2 (a− b). We study the parameter set Ωbdd of (a, b) where the map has at least one bounded orbit and the smaller parameter set where all orbits are bounded. We determine all cases where Tab is purely periodic. We also determine special parameter values where Tab has every orbit bounded and contained in an invariant circle with an irrational rotation number; The invariant circles are piecewise unions of arcs of conic sections. These parameter values yield quasiperiodic solutions to the nonlinear difference operator.