How can I perform a bifurcation analysis for a dynamical system of ODEs that has time-varying terms?
For example consider the ODE: $$\frac{dx}{dt}=f(x)+\nu(t)$$ where $\nu(t)$ is an additive term that depends on time $t$.
Is it possible to compute the Jacobian matrix to determine whether the system is critical and to determine the nature of the bifurcation? How would one compute the Jacobian for such systems? What are other options to perform a bifurcation analysis of such a system?
This type of continuous dynamical system is non-autonomous. One can write equivalently $$ \frac{d}{dt} \left( \begin{array}{c} x\\ \tau \end{array} \right) = \left( \begin{array}{c} f(x) + \nu(\tau)\\ 1 \end{array} \right) , $$ which is of the form $d y/dt = g(y)$ with $y = (x,\tau)^\top$. In the case where $\nu$ is a periodic function, the Melnikov method is a dedicated tool for this type of system.