I have the following vector PDE: $$ {\dot {\mathbf {x} }}(t)=\mathbf {A_{i(\gamma,x(t))}} \mathbf {x} (t)+\mathbf {B_{i(\gamma,x(t))}}\mathbf {u}, $$ where $\mathbf{x(t)} \in \mathbb{R}^{n}$. Some of the switching actions between the matrices depend on a bounded variable $\gamma \in [a,b]$ ($i(\gamma)$) and some also on the $\mathbf{x(t)}$ vector ($i(\gamma,x(t))$). Some of the switching actions are fixed ($i$). In addition, $\mathbf{x(t)}$ is continuous and the individual matrix entries are constant. The maximum order of the PDE is 2.
My question is: How can I prove that multiple sets of solutions: $\gamma \in [a,b]$ and $\mathbf{x(0)} \in \mathbb{R}^{n}$ exist, so that $\mathbf{x(t)}$ is periodic ($\mathbf{x(0)}=\mathbf{x(T)}$)?