To rewrite the given system in polar coordinates. There is at least only limit cycle, and that if there are so many, they all have the similar period T ( a,b ) is to be proved. Also prove that for b=0 , there is only one limit cycle. Concept Introduction: Poincare - Bendixson theorem states that if a dynamical system in which x ˙ = f (x, y), y ˙ = g (x, y) is continuously differentiable, and if it remains in a closed, bounded region R formed by trajectory of a vector field on the plane for all t ≤ 0 , then the trajectory must form a periodic orbit. Poincare - Bendixson Theorem states that if there are no fixed points in a closed, bounded region formed by the trajectory of a vector field on the plane, then trajectory must eventually approach a closed orbit. If there are no fixed points in the trapping region R, then according to the Poincare - Bendixson Theorem, R contains a closed orbit.

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Chapter 7.3, Problem 7E

Interpretation Introduction

Interpretation:

To rewrite the given system in polar coordinates. There is at least only limit cycle, and that if there are so many, they all have the similar period T(a,b) is to be proved. Also prove that for b=0, there is only one limit cycle.

Concept Introduction:

Poincare - Bendixson theorem states that if a dynamical system in which x˙ = f (x, y), y˙ = g (x, y) is continuously differentiable, and if it remains in a closed, bounded region R formed by trajectory of a vector field on the plane for all t≤0, then the trajectory must form a periodic orbit.

Poincare - Bendixson Theorem states that if there are no fixed points in a closed, bounded region formed by the trajectory of a vector field on the plane, then trajectory must eventually approach a closed orbit.

If there are no fixed points in the trapping region R, then according to the Poincare - Bendixson Theorem, R contains a closed orbit.

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