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Article Contents

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

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

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