Nonautonomous saddle-node bifurcations: random and deterministic forcing

TL;DR

This paper investigates how external random and deterministic forcing influence saddle-node bifurcations in interval maps, revealing the emergence of invariant graphs and strange non-chaotic attractors at critical parameters.

Contribution

It extends classical bifurcation theory to nonautonomous systems with forcing, introducing invariant graphs and analyzing their properties under various forcing conditions.

Findings

Existence of invariant graphs replacing fixed points under forcing.

At bifurcation, either a neutral invariant graph or a pair of pinched graphs occur.

In quasiperiodic forcing, pinched graphs are often strange non-chaotic attractors.

Abstract

We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. By replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both for random forcing by measure-preserving dynamical systems and for deterministic forcing by homeomorphisms of compact metric spaces. Additional assumptions like ergodicity or minimality of the forcing process then yield further information about the dynamics. The main difference to the unforced situation is that at the critical bifurcation parameter, two alternatives exist. In addition to the possibility of a unique neutral invariant graph, corresponding to a neutral fixed point, a pair of so-called pinched invariant graphs may occur. In quasiperiodically forced systems, these are often referred to as 'strange non-chaotic attractors'. The results on deterministic forcing can be considered as an extension of the work of Novo, Nunez, Obaya and Sanz on nonautonomous convex scalar differential equations. As a by-product, we also give a generalisation of a result by Sturman and Stark on the structure of minimal sets in forced systems.