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Analytical approximation for period-doubling following a Hopf bifurcation. (English) Zbl 0684.58030
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 $\mu\sp*$, 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 $\mu\sp*$ corresponds to the change of stability of the limit cycle (and to the disintegration of the center manifold).

37G99Local and nonlocal bifurcation theory
34C25Periodic solutions of ODE
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