Chapter 8.2, Problem 12E

Interpretation Introduction

Interpretation:

To calculate $\text{a }$ for the system ${\dot{\text{x}}\text{ = - y + xy}}^{\text{2}}$, ${\dot{\text{y}}\text{ = x + x}}^{\text{2}}$. If $\text{a < 0}$, the bifurcation is supercritical and if $\text{a > 0}$, 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 $\text{μ}$. If the decay becomes slower and slower and finally changes to growth at a critical value ${\text{μ}}_c$, the equilibrium state will lose stability. Then we say that the system has undergone a supercritical Hopf bifurcation.

A subcritical Hopf bifurcation occurs at $\text{μ = 0}$, where the unstable cycle shrinks to zero amplitude and engulfs the origin, rendering it unstable. For $\text{μ >0}$, 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.