Chapter 3.6, Problem 2E

Interpretation Introduction

Interpretation:

To find the effect of h on bifurcation diagram, where h is the imperfection parameter.

Plot the bifurcation diagram for ${\dot{\text{x}}\text{ = h + rx - x}}^{\text{2}}$, for $\text{h < 0}$, $\text{h = 0}$ and $\text{h > 0}$. Also sketch regions in the $\text{(r, h)}$ plane that correspond to qualitative different vector fields. Identify the bifurcation that occurs on the boundaries of those regions. Also plot the potential $\text{V(x)}$ corresponding to all different regions.

Concept Introduction:

General expression of the imperfect bifurcation system is $\dot{\text{x}}\text{ = h + rx - f}\left(\text{x}\right)$.

Here, $\text{x}$ is the position, $\text{h}$ is the imperfection parameter. and $\text{r}$ is a constant parameter.