Chapter 3.5, Problem 8E

Interpretation Introduction

Interpretation:

To show that ${\dot{\text{u}}\text{=au+bu}}^3-c{u}^5$ can be rewritten as $\frac{\text{dx}}{\text{dτ}}{\text{ = rx + x}}^{\text{3}}{\text{ - x}}^{\text{5}}$, where $a=0$.

Concept Introduction:

The supercritical pitchforkis similarlylabelled as forward bifurcation and is closely related to a continuous or second-order phase transition.

In statistical mechanics, the subcritical bifurcation is sometime called inverted or backward bifurcation and is related to discontinuous or first order phase transition.

The given first order system equation is ${\dot{\text{u}}\text{=au+bu}}^3-c{u}^5$ and has a subcritical pitchfork bifurcation at $a=0$.