Chapter 7.1, Problem 8E

Interpretation Introduction

Interpretation:

To find and classify all the fixed points of the system $\ddot{\text{x}}\text{ + a}\dot{\text{x}}\left({\text{x}}^{\text{2}}{\text{ + }\dot{\text{x}}}^{\text{2}}\text{ - 1}\right)\text{ + x = 0}$. To show that the system has a circular limit cycle, and to find amplitude and period of system. To find the stability of the limit cycle. To give an argument which shows that the limit cycle is unique, i.e., there are no other periodic trajectories.

Concept Introduction:

Fixed points occur where $\dot{\text{x}}\text{ = 0}$ and $\dot{\text{y}}\text{ = 0}$.

The Jacobian matrix at a general point $\left(\text{x, y}\right)$ is given by

$\text{A = }\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$

The Eigen value $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$

The solution for the quadratic equation can be calculated by $\text{λ = }\frac{\text{-b ± }\sqrt{{\text{b}}^{\text{2}}\text{- 4ac}}}{\text{2a}}$