Persistence of fixed points under perturbation in dynamical systems

Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs). Assuming that the system has a finite set of fixed points, I am interested in knowing (or obtaining references about) what is the behaviour of its fixed point structure under perturbations of the ODEs. More specifically, i would like to know under which conditions the total number of fixed points remains the same. By perturbations I mean generic changes in the system of equations... the more general the better.

I am sorry if the question is too basic, my interest comes from the study of the so-called renormalization flow in field theories. In particular, it would be important for me to generate an intuition about the conditions under which the approximations performed over a dynamical system alters its fixed point structure.

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It seems like the Conley index (en.wikipedia.org/wiki/Conley_index_theory) should have something to say about this, but I can't find any obvious references. – Steve Huntsman Jun 15 '10 at 17:01

5 Answers

A good reference for this sort of thing is Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983, if you're not already familiar with it (and even if you are, for that matter). Chapter 3 in particular is relevant to your question.

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A little bit more general than the previous answer: Given a singularity of a vector field such that the derivative is invertible, the singularity persists under perturbations. This is because the zero of the vector field in this case is transversal (so, it persist by C1 perturbations).

Otherwise, depending on the local structure in a local neighborhood, it can be "removed" after perturbation (in the case the index is zero) or bifurcate in more than one singularity (in the case the index is non zero and the singularity is not hyperbolic).

Generic singularities are hyperbolic in the dissipative setting. In the conservative one, one may have elliptic singularities robustly, but generically these will be also persistent.

This being local, may be studied in Rn as well as in a manifold.

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Aga, right! But if the fixed point is not hyperbolic then the local phase portrait may be destroyed. Though Federico didn't ask for it. – Andrey Gogolev Jun 15 '10 at 17:28

System of ODEs is a vector field. If there's finite number of fixed points and periodic orbits and the system is Morse-Smale then it is structurally stabel under $C^1$ small perturbations. In particular, all fixed points survive.

More details are here http://www.scholarpedia.org/article/Morse-Smale_systems

The manifold needs to be compact. So one extends $R^n$ to $S^n$ by adding an extra fixed point.

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If you are just interested in the number of the fixed points of the flow, you can just put your question in terms of zeros of the vector field; in this case the right tool is the topological degree, which is stable under small perturbations of the field in the uniform norm; and in absolute value is (generically) a lower bound on the number of the zeros. Of course, if the vector field is variational (it's the gradient if a functional) much stronger invariants are available (all the Morse complex machinery).

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In order to develop your intuition, you might want to start with a flow having no fixed points -- so a constant nonzero vector field, say in the plane. Perturb it by a nice vector field, eg. polynomial (or polynomial times a positive function decaying to zero at infinity) until you get zeros for your vector field. Now, what is it you want to show, or preserve? Do you want to estimate how big the perturbation need be (in terms of the size of the initial nonzero vector field) in order for zeros to be created?

I think you need to be a bit more specific about where you are headed, or what restrictions there are on your initial vector field and its allowed perturbations to get anywhere here.

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