Chapter 3.4, Problem 15E

Interpretation Introduction

Interpretation:

For the system ${\dot{\text{x}}\text{ = rx + x}}^{\text{3}}{\text{ - x}}^{\text{5}}$, the value of ${\text{r}}_{\text{c}}$ is to be calculated, where ${\text{r}}_{\text{c}}$ is defined by the condition that $\text{V}$ has three equally deep wells, i.e., the values of $\text{V}$ at the three local minima are equal.

Concept Introduction:

The potential is defined as $\dot{\text{x}}\text{ = f(x) = -}\frac{\text{dV}}{\text{dx}}$.

Critical points occur when $\frac{\text{dV}}{\text{dx}}=\text{ 0}$.

Local minima occur when $\frac{{\text{d}}^{\text{2}}\text{V}}{{\text{dx}}^{\text{2}}}\text{ > 0}$.