Hamiltonian system invariant in the unit circle but not periodic orbits

Question

I consider the a nonlinear system of the following:

\begin{align}\dot x&=x^2 + 3y^2 - 1 \\ \dot y &= -2xy \end{align}

I have to demonstrate that $x^2 + y^2 = 1$ is an invariant set, but not periodic orbits.

My attempt is that:

1. I first note that this system is Hamiltonian so that I can find a Hamiltonian function which is of the form: $$ H(x,y) = x^2 y + y^3 - y $$

2. I find that: if $(x,y) \in C = \{(x,y) \in \mathbb{R}^{2} | x^2 + y^2 = 1 \}$, then

$$ H(x,y) \equiv 0 $$

so $C$ is an invariant set.

But I am not really sure how to proceed to show that there exists no periodic orbits. Any hints will be much appreciated. Thanks in advance!.

Answer 1

You have correctly shown that the unit circle is invariant.

To show that it contains no periodic orbits, consider the equilibrium points of the system.