Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

${\text{V= (1- 4x}}^2-y^2{)}^2$.

Concept Introduction:

Fixed point of a differential equation is a point where, $\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as $\text{t}\to \infty$.