Chapter 8.4, Problem 9E

Interpretation Introduction

Interpretation:

The fixed points of the driven oscillations system satisfies ${\text{r}}^{\text{2}}\left({\text{ k}}^{\text{2}}{\left(\frac{\text{3}}{\text{4}}{\text{br}}^{\text{2}}\text{- a}\right)}^{\text{2}}\right)={\text{F}}^{\text{2}}$ is to be shown, and $\text{r vs a}$ for $\text{b = 0 and b }\ne \text{ 0}$ are to be plotted. This curve is single valued for small nonlinearity and triple-valued for large nonlinearity is to be shown, as well as explicit formula for ${\text{b}}_{{}_{\text{C}}}$ is to be found. Also, the cusp catastrophe surface yields when $\text{r}$ is plotted as a surface above $\text{(a,b)}$ are to be shown.

Concept Introduction:

The fixed points of the system equation are the points where $\text{r' = 0}$ and $\varphi \text{' = 0}$.

A catastrophe occurs for two control parameters and axis.

Catastrophe theory is a branch of bifurcation theory. It is used to study the dynamical systems.

It is the branch of bifurcation in which they study the dynamical behavior of the system equation in which the two $\text{-}$ control factor and one behaviors axis. The universal unfolding of the singularities is the cusp catastrophe and it has the system equation.