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

TL;DR

This paper investigates non-smooth saddle-node bifurcations in quasiperiodically driven differential equations, demonstrating the emergence of strange non-chaotic attractors and establishing the prevalence of such bifurcations in these systems.

Contribution

It extends the understanding of non-smooth bifurcations to continuous-time systems and shows their occurrence in a broad class of quasiperiodically driven differential equations.

Findings

Non-smooth saddle-node bifurcations lead to strange non-chaotic attractors.

Such bifurcations occur in a set with non-empty C2-interior within the parameter space.

The results connect discrete and continuous dynamical systems regarding bifurcation phenomena.

Abstract

Non-smooth saddle-node bifurcations give rise to minimal sets of interesting geometry built of so-called strange non-chaotic attractors. We show that certain families of quasiperiodically driven logistic differential equations undergo a non-smooth bifurcation. By a previous result on the occurrence of non-smooth bifurcations in forced discrete time dynamical systems, this yields that within the class of families of quasiperiodically driven differential equations, non-smooth saddle-node bifurcations occur in a set with non-empty C2-interior.