Chapter 8.5, Problem 2E

Interpretation Introduction

Interpretation:

For the given system ${\varphi }^{"}\text{+α}{\varphi }^{\text{'}}+\text{sin}\varphi \text{= I}$ to show that if $\text{α}$ is fixed and sufficiently small, the system’s stable limitcycle is destroyed in a homoclinic bifurcation as $\text{I}$ decreases. Show that if α is too large, the bifurcation is an infinite-periodbifurcation instead.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.