I'm studying a quasi-steady force model (published in a fluid dynamics journal) that consists of coupled, nonlinear ODEs that describe unsteady aerodynamics -- recently, my advisor and I have found some solutions to the system that appear to be stable-ish.
My question is: how can I proceed to confirm the stability of such solutions? I'm having a hard time thinking of a way to linearize the models for lift and drag forces, and then setting up an eigenvalue problem.
One way I've thought of is to manually perturb the state variables. That is, write functions that stop my solver when a steady state solution is detected, look at the final values of all of the state variables, then proceed to make a 2nd call to the solver, using these final values as the new initial values. And, changing any of the initial values before making the 2nd call to the solver is then effectively perturbing a state variable from its steady state. If after this small perturbation, the solutions gotten from the 2nd call to the solver again settle back into the same set of solutions (i.e. a steady-state solution for a fixed set of parameters), then this would indicate that the solutions could be stable.
Is this a good approach?
If so, should I manually perturb all of the state variables -- or only the state variables that I am interested in focusing on?
By what percentage should I perturb the state variables, in order to be able to conclude that the solutions are stable? Is there a common benchmark / standard to follow in perturbation / stability theory?
If this is not a good approach, can you recommend another approach?