Suppose that $\alpha, \beta, \gamma, \rho$ and $\kappa$ are non-negative constants and define $f(t) \geq 0$ to be some forcing function. I have been struggling to get some insight into the following non-linear, first-order, system of differential equations. \begin{align*} x'(t) & = f(t) + \alpha \, y(t) - \beta \, x(t), \\ y'(t) & = \gamma \, x(t) - \alpha \, y(t) + \rho \,y(t)\left(1-\frac{y(t)}{\kappa}\right). \end{align*} So far, I can show that if $f(t) = a \geq 0$, a stable asymptotic equilibrium point exists. Otherwise, I have been looking at $f(t)=\sin(t)$. Plots show that the system approaches a stable periodic orbit but I am not sure how to show this mathematically. Obviously, if there is a way to obtain a closed-form expression for $\left(x(t),y(t)\right)$ that would be ideal but I am not sure that's possible.
If anyone could provide some insight into how to show that as $t \to \infty$, the system for $f(t)=\sin(t)$ tends to a periodic orbit I would be extremely grateful. Thank you so much for your time.
Think of this system as a linear system $(\rho = 0)$. On those conditions, the stability is governed by the eigenvalues of $M = \left( \begin{array}{cc} -\beta & \alpha \\ \gamma & -\alpha \\ \end{array} \right)$ which are $\left\{-\frac{1}{2} \left(\sqrt{(\alpha -\beta )^2+4 \alpha \gamma }+\alpha +\beta \right),\frac{1}{2} \left(\sqrt{(\alpha -\beta )^2+4 \alpha \gamma }-\alpha -\beta \right)\right\}$. Now if for instance $\alpha=2,\beta = -1,\gamma = -\frac{17}{16}$ both eigenvalues are negative so the homogeneous response goes to zero remaining the forced response describing a periodic orbit after a transient. Incorporating now the nonlinear term $\rho y\left(1-\frac{y}{\kappa}\right)$ the linear theory doesn't help. Perhaps for very small $\rho$. Concluding, this system is nonlinear and depends on $\alpha,\beta, \gamma,\rho,\kappa$ parameters. General considerations are hard to establish. The best way to study is dividing to conquer.