Proving the existence of a center or a focus in a system of ODEs

Question

I'm asked to prove that for any odd value of $n\in\mathbb{N}$ there exist homogeneous polynomials $P_n,Q_n\in\mathbb{R}[x,y]$ such that the point $C=(0,0)\in\mathbb{R}^2$ is either a center or a focus of the system of ODEs $$\dot{x}=P_n(x,y),\quad \dot{y}=Q_n(x,y)$$ Here's my approach: I've shown that the origin is the only possible isolated critical point of the system. In this kind of problems, we usually consider the linear part of the system, which in this case would be the system $$\dot{x}=\langle\nabla P_n,(x,y)\rangle,\quad\dot{y}=\langle\nabla Q_n,(x,y)\rangle $$ in the origin. We would like to show that $C$ is a center or a focus of this new system. However, this only seems possible when $n=1$, since all the partial derivatives of $P_n$ and $Q_n$ are $0$ when evaluated at the origin if $n>1$, so I don't know how to proceed. Any help? Thank you in advance.

Answer 1

The question asks us to show the existence of such polynomials. Hence, we simplify the problem to the following system

$$\dot{x}=P(y)$$ $$\dot{y}=-Q(x).$$

We assume that $P$ and $Q$ only consist of odd powers. Now, we define the Lyapunov candidate function

$$V(x,y)=\int_{0}^{x}Q(\bar{x})d\bar{x} + \int_{0}^{y}P(\bar{y})d\bar{y}.$$

$P$ and $Q$ are odd the integrals are even, hence $V(x,y)>0$ for all $x,y \in \mathbb{R}-\{(0,0)\}$. For $\dot{V}$ we have by the fundamental theorem of calculus. $$\dot{V}=Q(x)\dot{x}+P(y)\dot{y}=Q(x)P(y)-P(y)Q(x)\equiv 0.$$ Hence, the origin is a center. Which can be seen as a constructive proof for the claim with the center.

Now, to the second part. We simplify the initial system from the question to

$$\dot{x}=\alpha \,x^{2n-1}$$ $$\dot{y}=\alpha \,y^{2n-1}.$$

Then we construct a Lyapunov candidate function $$V(x,y)=1/2x^2+1/2y^2$$ $$\implies \dot{V}=x\dot{x}+y\dot{y}=\alpha(x^{2n}+y^{2n}).$$

Depending on the sign of $\alpha$ we can show that $\dot{V}<0$ or $\dot{V}>0$. This is equivalent to showing that the origin is an asymptotically stable or unstable focus.