Chapter 8.3, Problem 2E

Interpretation Introduction

Interpretation:

To show that all trajectories eventually enter a certain trapping region for the system ${\dot{\text{x}}\text{ = a - x + x}}^{\text{2}}{\text{y, }\dot{\text{y}}\text{ = b - x}}^{\text{2}}\text{y}$, where $\text{a, b > 0}$. The system has a unique fixed point and to classify it. To show that the system undergoes a Hopf bifurcation when $\text{b - a = }{\left(\text{a + b}\right)}^{\text{3}}$ is to be shown. The Hopf bifurcation issupercriticalor subcritical is to be found. To plot the stability diagram in $\text{a, b}$ space.

Concept Introduction:

Nullclines are where $\dot{\text{x}}\text{ = 0}$, $\dot{\text{y}}\text{ = 0}$.

The Poincaré–Bendixson theorem:A trajectory must eventually approach a closed orbit if it is confined to a closed, bounded region that holds no fixed points.

Fixed points are the intersection of nullclines.

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

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

A Hopf bifurcation can only occur if the trace of the linearized system is zero.