Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 8.2, Problem 12E
Interpretation Introduction

Interpretation:

To calculate for the system x˙ = - y + xy2, y˙ = x + x2. If a < 0, the bifurcation is supercritical and if a > 0, the bifurcation is subcritical.

Concept Introduction:

Suppose we have a physical system that settles down to equilibrium through exponentially damped oscillations. Now suppose that the decay rate depends on a control parameter μ. If the decay becomes slower and slower and finally changes to growth at a critical value μc, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at μ = 0, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For μ >0, the large-amplitude limit cycle is suddenly the only attractor in town. Solution that used to remain near the origin is now forced to grow into large-amplitude oscillation.

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