(a) To sketch all the qualitatively different phase portraits arising due to varying parameters - ω 1 , ω 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 f(x) = 0 ; while substitution f(x) = x ˙ 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.

Question

Prove that a relation X defined on a set A that is reflexive, symmetric and antisymmetric is an equivalence relation and determine the equivalence classes.

Answer 1

Question

Chapter 8.6, Problem 1E

Interpretation Introduction

Interpretation:

(a) To sketch all the qualitatively different phase portraits arising due to varying parameters -ω1, ω2

(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 f(x) = 0 ; while substitution f(x) = x˙ 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.

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