Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows

Alessandro Colombo, Mike R Jeffrey

Research output: Contribution to journalArticle

47 Citations (Scopus)
148 Downloads (Pure)

Abstract

A vector field is piece wise smooth if its value jumps across a hyper surface, and a two-fold singularity is a point where the flow is tangent to the hypersurface from both sides. Two-folds are generic in piecewise smooth systems of three or more dimensions. We derive the local dynamics of all possible two-folds in three dimensions, including nonlinear effects around certain bifurcations, finding that they admit a flow exhibiting chaotic but nondeterministic dynamics. In cases where the flow passes through the two-fold, upon reaching the singularity it is unique in neither forward nor backward time, meaning the causal link between inward and outward dynamics is severed. In one scenario this occurs recurrently. The resulting flow makes repeated, but nonperiodic, excursions from the singularity, whose path and amplitude is not determined by previous excursions. We show that this behavior is robust and has many of the properties associated with chaos. Local geometry reveals that the chaotic behavior can be eliminated by varying a single parameter: the angular jump of the vector field across the two-fold.
Original languageEnglish
Pages (from-to)423-451
Number of pages29
JournalSIAM Journal on Applied Dynamical Systems
Volume10
Issue number2
DOIs
Publication statusPublished - 2011

Fingerprint Dive into the research topics of 'Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows'. Together they form a unique fingerprint.

  • Projects

  • Cite this