For the system, bifurcation occurs at the origin when μ = 0 . The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined. 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 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 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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Chapter 8.2, Problem 7E

Interpretation Introduction

Interpretation:

For the system, bifurcation occurs at the origin when $μ = 0$ . The phase portrait is to be plotted and the bifurcation is subcritical or supercritical is to be determined.

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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