Chapter 8.4, Problem 4E

Interpretation Introduction

Interpretation:

To explore the phase portrait of a system $\ddot{\text{θ}}+\left(1-\text{μ cosθ}\right)\dot{\text{θ}}+\text{sinθ}=0$ using a computer for $\mu \ge 0$. Find the value of $\mu$ for that system having stable limit cycle and state which type of bifurcation is creating and destroying the cycles as the value of $\mu$ greater than zero.

Concept Introduction:

• ➢ For two-dimensional system, limit cycles are created or destroyed by four ways:

  • 1) Hopf bifurcation

  • 2) Saddle-node or fold bifurcation of cycles: In this bifurcation, two limit cycles coalesce and annihilate.

  • 3) Infinite period bifurcation: In this bifurcation, the fixed points appear on the circle.

  • 4) Homoclinic or saddle-loop bifurcation: In this bifurcation, the limit cycle touches the saddle point and becomes a Homoclinic orbit. This is another type of infinite-period bifurcation.