Hot to show that system of nonlinear differential equations doesn't have periodic solutions?

Question

Suppose we have nonlinear system of differential equations $$ \frac{d\mathbf x}{dt} = \hat{A}(\mathbf x, \mu)\mathbf x $$ How to show that it has periodic solutions?

Answer 1

To prove or disprove the existence of periodic orbits in general $n$-dimensional nonlinear dynamical systems is a daunting task; the approach highly depends on the structure of the specific system. There is no general theory.

However, when the system is Hamiltonian (which is a major assumption and introduced a lot of structure), there is work done on the Weinstein conjecture, which is highly nontrivial, even in low-dimensional cases.