The system I am working on is in the following form: $$ \dot{x}=f(t,x,a)=f(t+T,x,a), $$ where $f(t,x,a) \in R^2$ is nonlinear and periodic in $t$, $T$ is a known constant, and $a$ is a vector of constant real parameters which can be optimized to make sure this dynamical system has a stable periodic solution. The detailed expression of the differential equations is : $$ \dot{x}_1=x_2 \\ \dot{x}_2=-\frac{1}{A(x_1,t)}[B(x_1,t)\ddot{\alpha}_1(t)+C(x_1,t)\ddot{\alpha}_2(t)+D(x_1,t)+E(x_1,t)+F(x1,x_2,t)] $$ where $$ A(x_1,t)=A_1-A_2cos(x_1-\alpha_1(t))+A_3 (\mbox{known to be always}>0),\\ B(x_1,t)=B_1cos(x_1-\alpha_1(t))+B_2,\\ C(x_1,t)=C_1cos(x_1-\alpha_2(t))+C_2,\\ D(x_1,t)=D_1sin(x_1-\alpha_1(t))\dot{\alpha}_1(t)^2+D_2sin(x_1-\alpha_2(t))\dot{\alpha}_2(t)^2+D_3sin(x_1),\\ E(x_1,x_2,t)=E_1sin(x_1-\alpha_1(t))x_2^2+E_2sin(\alpha_1(t)),\\ F(x_1,x_2,t)=F_1sin(x_1-\alpha_2(t))x_2^2+F_2sin(\alpha_2(t)),\\ $$ with constant parameters $A_1$, $A_2$, $A_3$, $B_1$, $B_2$, $C_1$, $C_2$, $D_1$, $D_2$, $E_1$, $E_2$, $F_1$ and $F_2$, and known periodic functions $\alpha_1(t)=\alpha_1(t+T_1)$ and $\alpha_1(t)=\alpha_1(t+T_2)$. $\alpha_1(t)$ and $\alpha_2(t)$ are parameterized by the vector $a$ and thus can be tuned (including $T_1$ and $T_2$) as well.
My current approach is: first, tune the parameters $a$ to find a periodic solution of the original nonlinear system; second, linearize it around the periodic solution and obtain a linear periodic system; finally, check the norm of the Floquent multipliers of the linearized system and decide if the periodic solution is stable.
I think I know how to check the stability once I find a periodic solution. The problem is, it seems very difficult to find a periodic solution to begin with. How to identify a periodic solution except for by directly looking at its time plot? The vector $a$ contains several parameters to be tuned, including the initial conditions. It doesn't seem feasible to check the plot of the solution corresponding to each $a$ because the feasible set of $a$ is large. Are there other ways to identify a periodic solution and its minimum period?
Any suggestion would be helpful to me. Let me know if there is anything not clear in the question description. Thank you!