I tried to solve an equation of Van-der-Pole using linearizations in the current point of the phase space (state-space). This method produced damped oscillations whereas the nonlinear model produced stable oscillations. Is there a method to obtain a solution closed to exact using linear models with some kind of linearizations of something like that? I would like to have a deal with eigenvalues and aigenvectors of the linear equations produced by the former nonlinear equations.
You can use RCAM, which reduces nonlinear equations to system of linear equations and focus on finding exact solutions. For detail, please follow Article An efficient scheme for accurate closed-form approximate sol...
ode45 matlab can visualize the evolution of the solutions
https://www.youtube.com/watch?v=hYEagJkfPMY&t=2s
You can look the Nonstandard Finite Difference (NSFD) method developed by Ronald Mickens.
In is book « Nonstandard Finite difference Models of Differential Equations », Mickens proposed in chapter 6, section 6.5.5., pp. 157-159 a very good numerical method for Van-Der-Pole equation (see relation (6.5.60)).
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