Differential Equations | Part IA, 2005

The Cartesian coordinates (x,y)(x, y) of a point moving in R2\mathbb{R}^{2} are governed by the system

dxdt=y+x(1x2y2),dydt=x+y(1x2y2).\begin{aligned} &\frac{d x}{d t}=-y+x\left(1-x^{2}-y^{2}\right), \\ &\frac{d y}{d t}=x+y\left(1-x^{2}-y^{2}\right) . \end{aligned}

Transform this system of equations to polar coordinates (r,θ)(r, \theta) and hence find all periodic solutions (i.e., closed trajectories) which satisfy r=r= constant.

Discuss the large time behaviour of an arbitrary solution starting at initial point (x0,y0)=(r0cosθ0,r0sinθ0)\left(x_{0}, y_{0}\right)=\left(r_{0} \cos \theta_{0}, r_{0} \sin \theta_{0}\right). Summarize the motion using a phase plane diagram, and comment on the nature of any critical points.

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