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Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications.

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Related Experiment Video

Updated: May 29, 2025

Kinematic History of a Salient-recess Junction Explored through a Combined Approach of Field Data and Analog Sandbox Modeling
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Nonsmooth folds as tipping points.

D J W Simpson1

  • 1School of Mathematical and Computational Sciences, Massey University, Palmerston North 4410, New Zealand.

Chaos (Woodbury, N.Y.)
|February 5, 2025
PubMed
Summary
This summary is machine-generated.

Nonsmooth folds in dynamical systems cause solutions to collide and disappear. Our study shows that simplified models often lack stable states after this bifurcation, leading to unpredictable system behavior.

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Area of Science:

  • Dynamical Systems Theory
  • Bifurcation Analysis
  • Nonsmooth Mechanics

Background:

  • Nonsmooth dynamical systems exhibit complex behaviors when solutions interact with switching manifolds.
  • Nonsmooth folds, characterized by the annihilation of solutions, represent a critical phenomenon in these systems.

Purpose of the Study:

  • To investigate the impact of nonsmooth folds on the existence of bounded invariant sets in dynamical systems.
  • To analyze the behavior of simplified models (truncated systems) following such bifurcations.

Main Methods:

  • Mathematical analysis of boundary equilibrium bifurcations in Filippov systems, hybrid systems, and continuous piecewise-smooth ordinary differential equations.
  • Examination of grazing-type events using continuous piecewise-linear maps.
  • Demonstration with a specific example to illustrate the absence of local invariant sets.

Main Results:

  • Leading-order truncations of systems experiencing nonsmooth folds generally lack bounded invariant sets beyond the bifurcation.
  • This absence of invariant sets is proven for various classes of nonsmooth systems and grazing events.
  • Higher-order terms are unlikely to restore local invariant sets, indicating a fundamental change in system dynamics.

Conclusions:

  • Nonsmooth folds can destabilize systems, causing attractors to shift to new states.
  • The findings provide insights into the global bifurcation structures observed in truncated nonsmooth dynamical systems.