Fixed-point densities for a quasiperiodic kicked-oscillator map

Suppose that a one-dimensional harmonic oscillator is subjected to instantaneous kicks q times per natural period, with the kick amplitude varying sinusoidally with position. Viewed stroboscopically in phase space, the motion has an infinitely extended periodic or quasiperiodic array of fixed points, as well as an infinite web of chaotic orbits. In the present work (restricted to the quasiperiodic case q=5) the fixed points are classified according to their local linear behavior, which depends essentially on a single variable, the residue R. With the aid of a five-dimensional embedding, a function rho(R) is calculated which for infinitesimal DeltaR gives the average density of fixed points in the plane with residue in the range (R,R+DeltaR). The location and strength of the singularities and discontinuities of rho(R) are extracted from relatively simple transcendental equations, and this makes possible efficient numerical determination of rho(R). An exact equality for the densities of positive-R and negative-R fixed points is proved using decagonal symmetry and the integral representation of rho(R). For parameter values below the period-doubling threshold, there are no unstable fixed points with R greater, similar 0, and so we have equality of the densities of stable centers and unstable saddles. (c) 1995 American Institute of Physics.