2010, 14(2): 739-776. doi: 10.3934/dcdsb.2010.14.739

Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach

1. 

Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85758 Garching, Germany

Received  September 2009 Revised  December 2009 Published  June 2010

We investigate local bifurcation properties for nonautonomous difference and ordinary differential equations. Extending a well-established autonomous theory, due to our arbitrary time dependence, equilibria or periodic solutions typically do not exist and are replaced by bounded complete solutions as possible bifurcating objects.
   Under this premise, appropriate exponential dichotomies in the variational equation along a nonhyperbolic solution on both time axes provide the necessary Fredholm theory in order to employ a Lyapunov-Schmidt reduction. Among other results, this yields nonautonomous versions of the classical fold, transcritical and pitchfork bifurcation patterns.
Citation: Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
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