Bifurcation of Periodic Delay Differential Equations at Points of 1:4 Resonance

Details Bifurcation of Periodic Delay Differential Equations at Points of 1:4 Resonance Bifurcation of Periodic Delay Differential Equations at Points of 1:4 Resonance

2010-01-08

arxiv.org/abs/1001.1193v1

Functional Differential Equations, 2006, vol. 13, Nr. 3-4, pp 519-536

Mathematics

Dynamical Systems

20 pages

Scientific paper

The time-periodic scalar delay differential equation $\dot x(t)=\gamma f(t,x(t-1))$ is considered, which leads to a resonant bifurcation of the equilibrium at critical values of the parameter. Using Floquet theory, spectral projection and center manifold reduction, we give conditions for the stability properties of the bifurcating invariant curves and four-periodic orbits. The coefficients of the third order normal form are derived explicitly. We show that the 1:4 resonance has no effect on equations of the form $\dot z(t)=-\gamma r(t)g(x(t-1))$.

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