Non-smooth saddle-node bifurcations 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. IntroductionBifurcation theory investigates qualitative changes in the long-term behaviour of a dynamical system along a continuous variation of the system. For the simple case of a monotonously increasing interval map, the dynamics are qualitatively understood if the fixed points of the respective system are known. Hence in this case bifurcation theory investigates the bifurcation of fixed points. A well-known example is the saddle-node bifurcation of a one-parameter family of concave functions: if the considered parameter is 'small', we have one attracting and one repelling fixed point, which approach each other upon the growth of the parameter until some threshold is reached. Above this threshold, these two fixed points have vanished. At the threshold itself, the two points merge together to one neutral fixed point.In general, non-autonomous systems ask for objects other than fixed points (which might not even exist) to describe the qualitative dynamics [22]. In the context of quasiperiodically forced monotone maps, a natural choice are invariant graphs. Like fixed points of monotone interval maps, these are barriers which can't be crossed by an orbit.

Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

“…We say that (θ, x) verifies (B1) n if (B1) n x ∈ C and θ Z − n−1 . [22,Lemma 4.8]). Let f ∈ V ω and assume (θ, x) verifies (B1) n for n ∈ N. Let 0 < L 1 < .…”

“…In this setting, Young [16] and Bjerklöv [17,18] developed powerful methods-in the spirit of the multiscale analysis and parameter exclusion techniques by Benedicks and Carleson [19]-to examine the occurrence and properties of SNA's. These methods had later been adapted to non-linear systems (such as ( * )) in [20,21,22,6].…”

Non-smooth saddle-node bifurcations III: Strange attractors in continuous time

“…Theorem 1.11 (cf. [14]). Let X ⊆ R be a non-degenerate interval, suppose ω ∈ T d is Diophantine and consider the space of one-parameter families…”

“…In the following, we will provide these estimates already adapted to the first return maps (Ξ β ) β∈[0,1] and concurrently prove that they are actually verified in the present situation. Note that there are subtle differences to the presentation in [14]. The reader interested in the details may consult [13].…”

Abundance of Mode-Locking for Quasiperiodically Forced Circle Maps

“…Yet, there is one well-established method of choice for the analysis of qpf circle diffeomorphisms in the hyperbolic regime -characterised by non-vanishing Lyapunov exponents -which is multiscale analysis and parameter exclusion in the spirit of Benedicks and Carleson [12]. In the above context, it was first developed by Young [13] and Bjerklöv [14,15] for the linear-projective case and later adapted to non-linear systems in [16,17]. Originally, this method was used to show the non-uniform hyperbolicity of certain quasiperiodic SL(2, R)-cocycles [13,14], which corresponds to the existence of strange non-chaotic attractors in the general case.…”