(Solved): Consider the system x=y+ax(x2+y2)y=x+ay(x2+y2) 2 (a) Linearize this system around the ...

Consider the system $$\begin{array}{c}\stackrel{?}{x}=?y+ax\left(x^2+y^2\right)\\ \stackrel{?}{y}=x+ay\left(x^2+y^2\right)\end{array}$$ $$2$$ (a) Linearize this system around the fixed point $$\left(x^{?},y^{?}\right)=(0,0)$$ and show that this predicts that the origin is a center (a fixed point surrounded by closed orbits). (a) Linearize this system around the fixed point $$\left(x^{?},y^{?}\right)=(0,0)$$ and show that this predicts that the origin is a center (a fixed point surrounded by closed orbits). (b) Next we want to move to polar coordinates. To this end, first show that $$\stackrel{?}{?}=\frac{x\stackrel{?}{y}?y\stackrel{?}{x}}{r^2}$$ (Hint: Use that $$\tan ?=y/x$$.) (c) Use the result from (b) and a similar equation for $$\stackrel{?}{r}$$ (which you can obtain from $$x^2+y^2=r^2)$$ to recast the differential system on polar coordinates. (d) Analyze the result in (c), showing that the actual behavior of the fixed point is that of a spiral, stable or unstable, depending on the value of $$a$$. This is because centers are very delicate; they require that trajectories close perfectly after each cycle, and even a slight miss converts them into spirals. Linear analysis is therefore not alway sufficient to show that a fixed point is a center, typically a conservation law or time reversal symmetry is required to ensure the existance of center fixed points.