The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations

TL;DR

This paper introduces a general mechanism for the creation of strange non-chaotic attractors during non-smooth saddle-node bifurcations in quasiperiodically forced systems, supported by simulations and rigorous proofs.

Contribution

It provides a rigorous description of non-smooth saddle-node bifurcations leading to SNA, including the concept of 'sink-source-orbits' and application to various parameter families.

Findings

Strange non-chaotic attractors arise during non-smooth saddle-node bifurcations.

The 'exponential evolution of peaks' characterizes pre-collision behavior.

Existence of SNA is proven via sink-source-orbits with positive Lyapunov exponents.

Abstract

We propose a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, which we call 'exponential evolution of peaks'. This observation is then used to give a rigorous description of non-smooth saddle-node bifurcations. The non-smoothness of the bifurcations and the resulting existence of SNA is established via the occurrence 'sink-source-orbits', meaning orbits with positive Lyapunov exponent both forwards and backwards in time. In order to demonstrate the flexibility of the approach, we discuss its application to some parameter families, including the so-called Harper map. Further, we prove the existence of strange non-chaotic attractors with a certain inherent symmetry, as they occur in non-smooth pitchfork bifurcations.