Chapter 8.6, Problem 1E

Interpretation Introduction

Interpretation:

• (a) To sketch all the qualitatively different phase portraits arising due to varying parameters -${\text{ω}}_{\text{1}}\text{, ω}{\text{2}}_{}$

• (b) To find the curves in the parameter space along which the bifurcation occurs.

• (c) To classify the bifurcations and to draw the stability diagram in the parameter space.

Concept Introduction:

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

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

• ➢ The diagram which shows the different types of behaviours that can occur in the parameter space is called the stability diagram.