Nonautonomous bifurcation of bounded solutions I: ALyapunov-Schmidt approach

Christian Pötzsche

doi:10.3934/dcdsb.2010.14.739

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2010, Volume 14, Issue 2: 739-776. Doi: 10.3934/dcdsb.2010.14.739

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

Christian Pötzsche^1,

1.
Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85758 Garching

Received: September 2009

Revised: December 2009

Published: September 2010

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Abstract

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.

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Mathematics Subject Classification: 37B55, 39A11, 34A34, 34C23, 37G99, 39A28, 47J15.

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