Proving the existence of periodicity for a system of ODEs

Question

The following system: $$\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} -y^{3} \\ x^3 \end{pmatrix},$$ can have periodic solutions but I'm not sure how to prove their existence. Usually I would linearise the system about it's fixed points to understand the behaviour of the system but it is not possible here. This is because the jacobian of the r.h.s evaluated at the fixed point of the system (0,0) gives a zeros matrix and resultantly, the eigenvalues are: $\lambda_1=\lambda_2=0$. Can the existence of periodic solutions be proved here or have I missed something that's obvious?

Answer 1

rewrite the coupled systems of two equations $$ \frac{dx}{dt} = -y^3 \\ \frac{dy}{dt} = x^3 $$ as $$\frac{dx}{dy} =- \frac{y^3 }{x^3} $$ and simplified to look $$x^3 \ dx + y^3 \, dy = 0. $$ you can integrate the exact differential to find the constant of motion to be the closed orbit $$x^4 + y^4 = \text{ constant} $$ indicating that the solution is periodic.

Answer 2

$$ x^4 + y^4 $$ is constant and a solution curve rotates around the origin...

$$ \frac{d}{dt} (x^4 + y^4) = 4 x^3 \dot{x} + 4 y^3 \dot{y} = 4 x^3(-y^3) + 4 y^3 (x^3) = 0 $$