I have a system of nonlinear ODEs that are of this form

$\displaystyle F(x)\ddot{x}+G(x,\dot{x})\dot{x}+Hx=R(x,\dot{x},u)$

where x are my states, F G and H are matrices (and also functions of the state and rate of change of state), and R is the forces on the system. u is a user input to the system.

So, I need to solve these equations. At the moment I'm using a time-marching method, which is good enough, but I'd like to explore other solutions that may solve some time. In particular I'm looking for a solution that may be very quick, and not necessarily accurate, but provide acceptable qualitative values that could be used to develop a control or optimisation algorithm for the system.

I've looked at potentially using asymptotic methods to approximate the solution, but I'm not sure how to apply it. The advantage of the particular application of these equations are that the frequency response will be very low, so the first approximation for the asymptotic method could be the solution where all rates of change of my system state are zero, and build from there.

Anyone with knowledge of asymptotic methods, or who has any ideas on the equation please message me. I'd be interested to hear your thoughts.