Chapter 8.4, Problem 12E

Interpretation Introduction

Interpretation:

The period of a closed orbit as a homoclinic bifurcation is approached is to be found by finding the time requires for a trajectory passing through $\left(\text{μ,1}\right)$ to escape from the saddle for a system ${\dot{\text{x}}\text{=λ}}_{\text{μ}}{\text{x,}\dot{\text{y}}\text{=-λ}}_{\text{s}}\text{y}$ is $\text{x(t)}\approx \text{1}\text{.}$

Concept Introduction:

• ➢ The solution of a system of two simultaneous first-order differential equations, if its matrix has two distinct real eigenvalues, is

  ${\text{x(t)=C}}_{\text{1}}{\text{k}}_1{\text{e}}^{{\text{λ}}_1\text{t}}{\text{ and y(t)=C}}_{\text{2}}{\text{k}}_2{\text{e}}^{{\text{λ}}_2\text{t}}$, where ${\text{λ}}_{\text{1,}}{\text{λ}}_{\text{2}}$ are eigenvalues and ${\text{k}}_1,{\text{k}}_2$ are eigen vectors.

• ➢ A limit cycle is a closed, isolated trajectory. A homoclinic bifurcation occurs when a limit cycle moves closer and closer to a saddle point.