Chapter 7.3, Problem 3E

Interpretation Introduction

Interpretation:

It is to be shown that the system $\dot{\text{x}}\text{=}{\text{x - y - x}}^{\text{3}}\text{, }\dot{\text{y}}\text{=}{\text{x + y - y}}^{\text{3}}$ has a periodic solution.

Concept Introduction:

Poincare $\text{-}$ Bendixson theorem states thatif a dynamical system in which $\dot{\text{x}}\text{ = f (x, y), }\dot{\text{y}}\text{ = g (x, y)}$ are continuously differentiable, and if it remains in a closed, bounded region $\text{R}$ formed by trajectory of a vector field on the plane for all $\text{t}\le \text{0}$, then the trajectory must form a periodic orbit.

Poincare $\text{-}$ Bendixson Theorem states that if there are no fixed points in a closed, bounded region formed by trajectory of a vector field on the plane, then trajectory must eventually approach a closed orbit.

If there are no fixed points in the trapping region R, then according to the Poincare $\text{-}$ Bendixson Theorem, R contains a closed orbit.