Loss of determinacy at small scales, with application to multiple time-scale and nonsmooth dynamics

S Webber, M Jeffrey*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review


A singularity is described that creates a forward time loss of determinacy in a two-timescale system, in the limit where the timescale separation is large. We describe how the situation can arise in a dynamical system of two fast variables and three slow variables or parameters, with weakly coupling between the fast variables. A wide set of initial conditions enters the ε-neighbourhood of the singularity, and explodes back out of it to fill a large region of phase space, all in finite time. The scenario has particular significance in application to piecewisesmooth systems, where it arises in the blow up of dynamics at a discontinuity and is followed by abrupt re-collapse of solutions to ‘hide’ the loss of determinacy, and yet leave behind a remnant of it in the global dynamics. This constitutes a generalization of a ‘micro-slip’ phenomenon found recently in spring-coupled blocks, whereby coupled oscillators undergo unpredictable stick-slipstick sequences instigated by a higher codimension form of the singularity. The indeterminacy is localised to brief slips events, but remains evident in the indeterminate sequencing of nearsimultaneous slips of multiple blocks.
Original languageEnglish
Article number2150041 (2021)
JournalInternational Journal of Bifurcation and Chaos
Issue number3
Publication statusPublished - 15 Mar 2021


  • discontinuity
  • sliding
  • sticking
  • friction
  • slip
  • switching
  • oscillators


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