Chapter 3.4, Problem 14E

Interpretation Introduction

Interpretation:

The algebraic expressions for all the fixed points of a system ${\dot{\text{x}}\text{ = rx + x}}^{\text{3}}{\text{- x}}^{\text{5}}$ exhibiting subcritical pitchfork bifurcation as r varies are to be determined. The vector fields as r varies are to be sketched including all fixed points and their stability and $r_s$ in saddle-node bifurcation should be calculated.

Concept Introduction:

The fixed points of the system occur at $\dot{\text{x}}\text{ = 0}$.

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

In pitchfork bifurcation, the fixed points appear and disappear in symmetrical pairs.

There are two types of pitchfork bifurcation, one is supercritical and another is subcritical.

In supercriticalbifurcation, a stable fixed point is present and after changing parameters it becomes unstable and two new symmetric unstable points generate.

In subcriticalbifurcation, an unstable fixed point is present and after changing parameters it becomes stable and two new symmetric stable points generate.

Saddle-node bifurcation is one of the bifurcation mechanism in which fixed points create, collide and destroy.