Chapter 7.2, Problem 10E

Interpretation Introduction

Interpretation:

To show thesystem ${\dot{\text{x}}\text{ = y - x}}^3{\text{, }\dot{\text{y}}\text{ = - x - y}}^3$ has no closed orbits by constructing a Liapunov function ${\text{V = ax}}^2{\text{ + by}}^2$ with the suitable $\text{a, b}$.

Concept Introduction:

Construct Liapunov function ${\text{V = ax}}^2{\text{ + by}}^2$

Find $\dot{\text{V}}$ by taking derivative.

Using system equation, determine $\dot{\text{V}}$ and $\text{V}$