Analytical approximation for period-doubling following a Hopf bifurcation (Q1825492)

This work is concerned with the behavior of a system of n autonomous ordinary differential equations of the form \(\dot x=F(x,\mu)\), where x and F(x,\(\mu)\) are n-vectors, and where \(\mu\) is a scalar parameter. It frequently happens that a change in \(\mu\) causes an equilibrium point to change its stability resulting in the birth of a periodic motion called a limit cycle, a process known as a Hopf bifurcation. If the change in \(\mu\) is then continued, the limit cycle may itself undergo a change of form via a period-doubling bifurcation. This scenario is often the first step in a sequence of period-doublings leading to chaos. The goal of this work is to suggest an analytic approach which will yield an approximation for \(\mu^*\), the critical value of \(\mu\) corresponding to the first period-doubling bifurcation. The idea of the method is to use center manifold theory to approximate the newly-born limit cycle, and then to use that approximation to investigate the stability of the limit cycle. The critical value \(\mu^*\) corresponds to the change of stability of the limit cycle (and to the disintegration of the center manifold).