To classify the fixed point at the origin of the system x ˙ = x - y - x ( x 2 + 5y 2 ) , y ˙ = x + y - y ( x 2 + y 2 ) . Using r r ˙ = x x ˙ + y y ˙ , and θ ˙ = ( x y ˙ - y x ˙ ) r 2 the given system should be rewritten in polar coordinates. The circle of maximum radius, r 1 and of minimum radius, r 2 , centered on the origin are to be determined. Also, To prove that the given system has a limit cycle somewhere in the trapping region r 1 ≤ r ≤ r 2 . Concept Introduction: The Jacobian matrix at a general point ( x, y ) is given by A = ( ∂ x ˙ ∂ x ∂ x ˙ ∂ y ∂ y ˙ ∂ x ∂ y ˙ ∂ y ) The eigenvalue λ can be calculated using the characteristic equation | ( A - λI ) | = 0 The solution of the quadratic equation is λ = -b ± b 2 - 4ac 2a The circle of maximum radius centered on the origin with all trajectories having radially outward component on it can be obtained by r ˙ ≥ 0 . The minimum radius circle centered on the origin with all trajectories having component directed radially inward on it can be found by putting r ˙ ≤ 0 . Nullclines are the curves in the phase portrait where r ˙ = 0 or θ ˙ = 0 . According to Poincare-Bendixson theorem, if a trapping region for a system does not contain any fixed point, then there must be at least one limit cycle within this trapping region.

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Chapter 7.3, Problem 1E

Interpretation Introduction

Interpretation:

To classify the fixed point at the origin of the system x˙ = x - y - x(x2 + 5y2), y˙ = x + y - y(x2 + y2). Using rr˙ = xx˙ + yy˙, and θ˙ = (xy˙ - yx˙)r2 the given system should be rewritten in polar coordinates. The circle of maximum radius, r1 and of minimum radius, r2, centered on the origin are to be determined. Also, To prove that the given system has a limit cycle somewhere in the trapping region r1≤r≤r2.

Concept Introduction:

The Jacobian matrix at a general point (x, y) is given by

A = (∂x˙∂x∂x˙∂y∂y˙∂x∂y˙∂y)

The eigenvalue λ can be calculated using the characteristic equation

|(A - λI)| = 0

The solution of the quadratic equation is λ = -b ± b2- 4ac2a

The circle of maximum radius centered on the origin with all trajectories having radially outward component on it can be obtained by r˙≥0.

The minimum radius circle centered on the origin with all trajectories having component directed radially inward on it can be found by putting r˙≤0.

Nullclines are the curves in the phase portrait where r˙ = 0 or θ˙ = 0.

According to Poincare-Bendixson theorem, if a trapping region for a system does not contain any fixed point, then there must be at least one limit cycle within this trapping region.

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