when I studied some delay systems, the fixed points of these systems have zero eigenvalue. I don't how can study its stability?
One way is to use desingularization of such point and to study the local stability near the couple of more regular points or to study behavior asymptotically
Dear Yakov Krasnov
can you recommend any source or paper explain this desingularization method?
Another way would be to use the center manifold theorem. To simplify, we assume 0 is the only imaginary root of the characteristic equation (other cases can be dealt with in a similar way but the calculations can be too difficult) and that all the other eigenvalues have strictly negative real part (otherwise the fixed point would be unstable).
The center manifold theorem consists of reducing the local stability analysis of any DDE to that of a system ODEs (with dimension being equal to the multiplicity of zero as a root of your characteristic equation).
The calculations of the center manifold can be tedious but doable. The resulting system ODEs may be difficult to analyse though.
Yes, I agree, but first take a look on "Qualitative Theory of Planar Differential Systems"
By Freddy Dumortier, Jaume Llibre, Joan C. Artés, Page 120, 3. Desingularization of Nonelementary Singularities
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