# Do periodic solutions of this system always exist?

Consider the following system of autonomous differential equations on $\mathbb{R}$:

$$\frac{d}{dt}x(t)=f(x(t),y(t)) \ \ \ \ \ \text{and} \ \ \ \ \frac{d}{dt}y(t)=g(x(t),y(t)) \ \ \ \ (1)$$ where $f$ and $g$ satisfy local Lipschitz condition on both $x$ and $y$.

While I was plotting some graphs modeling some ecological models, such as models of the interaction between populations of predators and preys, I noticed that this type of systems always have a periodic solution, provided it has a global bounded solution. Usually this periodic solution is a steady state (a constant solution).

Is there any theorem which states this result? That is, is this theroem correct:

Theorem. If $f$ and $g$ are locally Lipschitz with respect to both variables and if $(1)$ has a bounded solution on $t\in[0,\infty)$, then $(1)$ has a periodic solution.

If there is a bounded solution, its $\omega$-limit set is compact. Now use the Poincaré–Bendixson theorem.