I guess here I am not sure how to get started, I know the definitions:
The $ω$-limit sets of points are the set of points that the system of equation approach as time goes to infinity, and the $α$-limit sets of points are the points approached as t goes to negative infinity. But I am not sure how the eigenvalues or vectors help me determine the limit sets.
But I don't understand how to apply them.
My conclusion for both of these question is that $α(0)=ω(0)={0}$ because in both cases the origin is a fixed point. Is this correct? Are the more limit sets than these?
Consider the systems of equations and find all the $α$- and $ω$-limit sets of points in the plane.
a. $$\dot {x}=y+x(x^2+y^2-1)$$ $$\dot {y}=-x+y(x^2+y^2-1)$$
which in polar coordinates, is given by:
$$\dot {r}=r(r^2-1)$$ $$\dot {ø}=-1$$
$$r^2=x^2+y^2$$ $$r \dot {r}=x \dot {x}+y \dot {y}$$ $$=r^2(r^2-1)$$ $$r \dot=r(r-1)(r+1)$$
So I understand how the got the value for $\dot {r}$, but I don't understand how to get the value for $\dot {ø}$
The fixed points will be when $\dot {r}=0$ so when r=1, -1, 0.
From my book: "Because $\dot {ø}$=-1, thus the solution goes clockwise around the origin at unit angular speed." Why??
b. $$\dot {x}=y+x(1-x^2-y^2)(x^2+y^2-4)$$ $$\dot {y}=-x+y(1-x^2-y^2)(x^2+y^2-4)$$
which in polar coordinates, is given by:
$$\dot {r}=r(1-r^2)(r^2-4)$$ $$\dot {ø}=-1$$
