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

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.