I have a model that consist of two ODEs and one PDE which are all coupled and non-linear. When I linearize the system around the steady state and simulate the system with initial values that are different from steady state I get a stationary deviation in one of the solutions. Due to non-linearities I expect a quite different behavior for the linear system. However, if I multiply some the elements of the system matrix with a constant (relaxation or dampning coefficient?), the linear system now resembles the non-linear system much better in terms of transient behavior and that I no longer have a stationary deviation which I had before. Is this method for fitting the linear model allowed and have some theory that can justify the use of such relaxation constants? Or is it just a 'cheat' which is not valid as the system does change due to the use of such constant?
I would like to suggest you that please use homogeneous system of equations for system analysis. To do so, you need to linearize the nonlinear part and add it with the linear part of your system. Accordingly, you will get meaningful system for your data analysis.
Without knowing the equations, this sounds to me a little bit like Successive Over-Relaxation Method or Weighted Jacobi Method. You might want to look into these.
It is difficult to give any hint without knowing the equations. However, at a general level linearization around an equilibrium might not provide a correct behaviour if the linearized system is not exponentially stable, i.e., if some of its eigenvalues are zero. For example, the ODE
y'=-y3
is stable around y=0, but its linearization is
y'=0
for which each initial condition is stationary.
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