Interraption: To show that the system r˙=r(1−r2),θ˙=1 is equivalent to x˙=x−y−x(x2+y2),y˙=x+y−y(x2+y2) for polar to Cartesian coordinates. A limit cycle is a closed trajectory. Isolated means that neighboring trajectories are not closed. A limit cycle is said to be unstable or half stable, if all neighboring trajectories approach the lemin cycle. These systems oscillate even in the absence of external periodic force.

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rogreenhoxa8 · Accepted Answer

The coordinate system can be either the Cartesian system or the polar system. In case of the polar system, a point is determined by a distance and angle from a reference point while as in Cartesian coordinate system each point is determined by a pair of numerical coordinates. It is given that x=rcos⁡θandy=rsin⁡θ Square and add the above equations. x2+y2=r2cos2θ+r2sin2θ x2+y2=r2(cos2θ+sin2θ) Substitute (cos2θ+sin2θ)=1 so that x2+y2=r2 Now, x=rcos⁡θ Differentiate. x˙=r˙θ−rθsin⁡θ Substitute r˙=r(1−r2),cos⁡θ=xrandy for rsin⁡θ into the above equation. x˙=(x(r(1−r2))r)−y x˙=x(1−r2)−y Substitute r2=x2+y2 x˙=x(1−x2−y2)−y x˙=x−y−x(x2+y2) Thus, it is shown that x˙=x−y−x(x2+y2) Now, y=sin⁡θ Differentiate. y˙=r˙sin⁡θ+rθcos⁡θ Substitute r˙=r(1−r2),sin⁡θ=yr,θ˙=1andx=rcos⁡θ into the above equation. y˙=(y(r(1−r2))r)+x y˙=y(1−r2)+x But r2=x2+y2 y˙=x+y−y(x2+y2) Therefore, it is shown that y˙=x+y−y(x2+y2) Answer: For system

Interraption:To show that the system \displaystyle\dot{{r}}

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