If we have a nonlinear dynamical system how are we supposed to perform parameter estimation, for different parameters and with respect to which output.
How are we going to come up with an optimization scheme, and choose which set of parameter to optimize the with respect to which output/s.
we need to find the correlation parameters between o/p and i/p, phase plots also help to determine dependencies.
You can either look for mathematical constraints (depending on the physical nature of the system you are looking into) that could help to limit the range of some of those parameters. Say for e.g. some parameter in your equation cannot be negative.
One simplest technique would be to vary each parameter while keeping others fixed and see how the output changes wrt to that parameter. Take for example a change of +-10 fold. This would help to check the sensitivity of your model's o/p with the parameters.
Sensitivity Analysis (hinted at by Chakit) will give you some idea of which parameters to prioritise, but the results will only be valid for narrow regions of operation.
Have a look at the linked article for an idea on how to perform parameter estimates over multiple uncertain outputs. The field (rather confusingly) called history matching is fairly recent and dedicated to this sort of work under extremely uncertain and data-scarce problems.
Article History matching for exploring and reducing climate model pa...
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