Chapter 8.3, Problem 1E

Interpretation Introduction

Interpretation:

For Brusselator model of a chemical oscillator ${\dot{\text{x}}\text{ = 1- (b +1)x + ax}}^{2}y$, ${\dot{\text{y}}\text{ = bx - ax}}^{2}y$, find all the fixed points using the Jacobian matrix and classify them. Sketch the nullclines and build a trapping region. A Hopf bifurcation occurs at some parameter value $b=b_c$ is to be shown, where $b_c$ is to be determined. Does the limit cycle exist for $b>b_c$ or $b<b_c$? The approximate period of the limit cycle for $b\approx b_c$ is to be determined.

Concept Introduction:

• ➢ The stability for every fixed point can be determined using Jacobian matrix.

• ➢ The trapping region can be constructed using limit cycle.

• ➢ Poincare-Bendixson theorem can be used to determine limit cycle.

• ➢ The period of limit cycle can be determined using Eigen values.