Is it correct to use the Jacobian matrix in determining the types of equlibria of a non-linear chaotic system of smooth ODEs? If not, is there a general approach?
2026-03-25 11:13:37.1774437217
Types of equilibria of a chaotic system
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Stability analysis of equilibrium points of non-linear systems can get quite difficult. If the Jacobi matrix has eigenvalues with real part bounded away from zero, then you're fine; the type of stability is of the non-linear system is the same as in the linearized case, however the structure might still be different. If the real part of one of the eigenvalues is zero, then even small non-linearities can change the stability of the point. In this case different techniques have to be applied. Unfortunately there is no general approach.