Chapter 7.2, Problem 24P

Theorem $\mathrm{7.2.2}$ provides no information about the stability of a critical point of an almost linear system if that point is a centre of the corresponding linear system. The systems

$\frac{dx}{dt}=y+x\left(x^2+y^2\right)$, (i)

$\frac{dy}{dt}=-x+y\left(x^2+y^2\right)$

and

$\frac{dx}{dt}=y-x\left(x^2+y^2\right)$, (ii)

$\frac{dy}{dt}=-x-y\left(x^2+y^2\right)$ $$

show that this must be so.

(a) Show that $\left(0,0\right)$ is a critical point of each system and, furthermore, is a centre of the corresponding linear system.

(b) Show that each system is almost linear.

(c) Let $r^2=x^2+y^2$, and note that $xdx/dt+ydy/dt=rdr/dt$. For system (ii), show that $dr/dt<0$ and that $r\to 0$ as $t\to \infty$; hence the critical point is asymptotically stable. For system (i), show that the solution of the initial value problem for $r$ with $r=r0$ at $t=0$ becomes unbounded as $t\to \frac{1}{2}{r}_0^2$; hence the critical point is unstable.