Chapter 6.3, Problem 9E

Interpretation Introduction

Interpretation:

To find all the fixed points and classify them for the system equation ${\dot{\text{x}}\text{ = y}}^3\text{- 4x }$, ${\dot{\text{y}}\text{ = y}}^{\text{3}}\text{- y - 3x }$. Show that the line $\text{x = y }$ is invariant. Show that $\left|\text{x(t) - y(t)}\right|\to \text{0}$ as $\text{t }\to \infty$ for all other trajectories. Sketch the phase portrait.

Concept Introduction:

Determine the fixed point of the system.

Use linearize system and determine the stability for every fixed point.

Represent the system equation in vector field on the phase plane.

Show the fixed points in phase portrait.