Chapter 3.4, Problem 9E

Interpretation Introduction

Interpretation:

The value of $r$ at which bifurcation occurs is to be determined and it should be classified for $\dot{\text{x}}\text{ = x + tanh(rx)}$. Also, the bifurcation diagram of fixed points ${\text{x}}^{*}\text{vs r}$ is to be sketched.

Concept Introduction:

The qualitative change in the dynamics of the flow with parameters is called bifurcation, and the points at which this occurs is called bifurcation points.

Bifurcation is used to study the stability of the dynamical systems.

By changing the parameter, the fixed points move towards each other, collide and mutually annihilate, known as Saddle Node Bifurcation.

The stabilities of the fixed points interchanged by changing the parameter is known as transcritical Bifurcation.

When the single stable fixed point is present and it turns to an unstable point due to change in parameter and two new symmetric stable fixed points occurs is called Supercritical Pitchfork Bifurcation.

When the single unstable fixed point is present and it turns to a stable point due to change in parameter and two new symmetric unstable fixed points occurs is called Subcritical Pitchfork Bifurcation.