Chapter 8.2, Problem 9E

Interpretation Introduction

Interpretation:

To sketch the nullclines and bifurcations that occur as parameter b varies.

To show that the positive fixed point exists for all $\text{a, b > 0}$

To show that Hopf bifurcation occurs at the positive fixed point if

${\text{a= a}}_{\text{c}}\text{=}\frac{\text{4(b-2)}}{{\text{b}}^{\text{2}}\text{(b+2)}}$

To check validity and plot the phase portrait of c.

Concept Introduction:

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

Nullclines are the curves where either $\dot{\text{x}}=0\text{ or }\dot{\text{y}}\text{ = 0}$. They show whether the flow is completely vertical or horizontal.

Hopf bifurcation is the point where system loses its stability and fixed solution turns into periodic one.

Phase portraits represent the trajectories of the system with respect to the parameters and give qualitative idea about evolution of the system, its fixed points, whether they will attract or repel the flow, etc.