A posteriori, one can always compare the two numerical solutions for the nonlinear and linear case, respectively. A priori, one could try use an estimate of

the largest eigenvalues of the Jacobian (larger positive eigenvalues will move the system away faster from the datum around which linearization was carried out). In principle, one could also check the norm of the Hessian matrices of the components of the right hand side (the bigger the Hessians, the stronger the nonlinear effects are going to be). However, this is far from general and easy.

First, we cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking at the qualitative dynamics of a system. Equilibrium points– steady states of the system– are an important feature that we look for. Many systems settle into a equilibrium state after some time, so they might tell us about the long-term behavior of the system.

Equilibrium points can be stable or unstable: put loosely, if you start near an equilibrium you might, over time, move closer (stable equilibrium) or away (unstable equilibrium) from the equilibrium. Physicists often draw pictures that look like hills and valleys: if you were to put a ball on a hill top and give it a push, it would roll down either side of the hill. If you were to put a ball at the bottom of a valley and push it, it would fall back to the bottom of the valley.

Dear Victor, as long as I know, for a nonlinear system you have no way of treating easily the problem in the frequency (or in the Laplace) domain. In my opinion, you should work on the time-domain representation of the system and approximate the evolution of both systems by a one-step numerical scheme (e.g., forward Euler in the simplest case), applying the same control. Then, you can compare the resulting trajectories of the original and of the linearized system.

in many cases the most important difference is not quantitative but the qualitative one. the principle of superposion does not work in nonlinear system and the new type solutions comparing to the linear one are present. the comparison is available only for the weakly nonlinear solutions.

Thanks much. I compared my trajectories producing different controllers (linear and nonlinear control) but I couldn't compare something like pure dynamics of the controllable object because I used linearization. And this linearization changed the controllable object and fetch other properties (for instance other bandwidth). How can I measure it? What the norm of Hessian show me? When I use my linearized system the Hessian equals zero. But for nonlinear its don't. And what does it mean? Also, I thought of using some function norms to compare my models. Maybe it helps me?

Try to using RK method and Euler modify method (numerical Solutions).

Also use to KBM method or Harmonic balance method for nonlinear

The Harmonic Balance method can be a very useful tool for such analysis. It is essentially an alternating time-frequency method, which uses both the direct and the inverse Fourier transforms back and forth. You can therefore use the time representation of the problem when performing nonlinear operations and project the solution back on a set of harmonic basis functions otherwise. If you would rather look at an algebraic system, then the linear components are on the diagonal (blocks) of that matrix, while the off-diagonal components represent the nonlinear cross-coupling terms.