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arXiv · 1307.0347

Non-Smooth Saddle-Node Bifurcations of Forced Monotone Interval Maps I: Existence of an SNA

Gabriel Fuhrmann
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Abstract

We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as "SNA". In contrast to former results in this direction, our conditions are C^2-open in the fibre maps. By applying a general result about saddle-node bifurcations in skew products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

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Gabriel Fuhrmann. 2013-08-06. Non-Smooth Saddle-Node Bifurcations of Forced Monotone Interval Maps I: Existence of an SNA. https://arxiv.org/abs/1307.0347

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