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

Maiclubk 2020-11-03 Answered

Interraption: To show that the system r˙=r(1r2),θ˙=1 is equivalent to x˙=xyx(x2+y2),y˙=x+yy(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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Expert Answer

rogreenhoxa8
Answered 2020-11-04 Author has 109 answers

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=rcosθandy=rsinθ
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=rcosθ
Differentiate.
x˙=r˙θrθsinθ
Substitute r˙=r(1r2),cosθ=xrandy for rsinθ into the above equation.
x˙=(x(r(1r2))r)y
x˙=x(1r2)y
Substitute r2=x2+y2
x˙=x(1x2y2)y
x˙=xyx(x2+y2)
Thus, it is shown that x˙=xyx(x2+y2)
Now,
y=sinθ
Differentiate.
y˙=r˙sinθ+rθcosθ
Substitute r˙=r(1r2),sinθ=yr,θ˙=1andx=rcosθ into the above equation.
y˙=(y(r(1r2))r)+x
y˙=y(1r2)+x
But r2=x2+y2
y˙=x+yy(x2+y2)
Therefore, it is shown that y˙=x+yy(x2+y2)
Answer:
For system

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