If you have a dynamical system with zero eigenvalues in the linear part, one common method to restrict it to the center manifold is to transform it such that the linear terms are in Jordan normal form and then apply transformations of the form y=h(x,..) to isolate the center manifold. h can be assumed as a partial power series to get an easy approximate solution. This is easily applied if the dynamical system has polynomial form and there are many examples on the web.
But how do you apply this method if the system contains non-polynomial terms? Can we use a Taylor-expansion? Has anyone a good step-by-step example?
Hello!
Jack Carr's book "Applications of Center Manifold Theory" is a good reference.
It is shown how to approximate center manifolds.
S. Naciri
I used the Taylor's series in a vector field which are not polinomial and the center manifold theory is the same
The question is more whether you want the center manifold or the dynamics on the center manifold. There is a good procedure for the latter, see for instance, Kuznetosv YA Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODEs. SIAM J. Numer. Anal. 36 (1999), 1104-1124 or his book.
Also this procedure relies on Taylor series which is fine if you do not have flat terms.
The theorem on the center manifold does not rely on any assumptions on the r.h.s.of the system under consideration (except for needed smoothness). You can use it to any system which conditions of the theorem hold for.
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