Chapter 8.2, Problem 10E

Interpretation Introduction

Interpretation:

Find fixed points of the systemand classify them. By considering nullclines, construct a trapping region. Find conditions on A, B, qfor which the system has a stable limit cycle.

Concept Introduction:

Nullclinesare a set of points at which $\dot{\text{x}}\text{ = 0 or }\dot{\text{y}}\text{ = 0}$. It represents whether the flow is purely horizontal or vertical.

To check the stability of fixed point, use Jacobian matrix

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

The point $(x^{*},y^{*})$ is said to be a stable fixed point if eigenvalues of Jacobian matrix evaluated at this point have negative real parts, and point is said to be unstable if one of its eigenvalues has positive real part. If both the eigenvalues are purely real, then the fixed point is saddle point.

The fixed points are stable if ${\tau }^2-4\Delta >0$ for Jacobian matrix at a fixed point.