EFFECT OF PERTURBATION ON THE AREA PRESERVATION MAP

Abstract

Subharmonic bifurcation patterns in area-preserving maps change their forms dramatically under dissipative perturbation. The studies of such changes are done by the use of the normal form theory and the Lyapunov-Schmidt reduction method where the eigenvalues of the linearized map move along a circle of radius (1 - epsilon)1/2 in the complex plane. The results are: (i) one set of three-cycle branches, which is hyperbolic, becomes separated from the origin under the dissipative perturbation; (ii) a pair of four-cycle branches, which can be hyperbolic for both branches, or hyperbolic for one branch and stable for the other one depending on the condition, is separated from the origin also under the same perturbation; (iii) one set of n-cycle (n greater-than-or-equal-to 5) branches, which is elliptic for the area-preserving case, is retained under the perturbation for sufficiently small epsilon.