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An efficient algorithm for solving piecewise-smooth dynamical systems.

Nicola Guglielmi1, Ernst Hairer2

  • 1Gran Sasso Science Institute, via Crispi 7, I-67100 L'Aquila, Italy.

Numerical Algorithms
|February 21, 2022
PubMed
Summary

This study introduces a novel algorithm for numerically treating piecewise-smooth dynamical systems. It selects a unique solution from non-unique cases, avoiding complex regularized differential equations.

Keywords:
Codimension-2 manifoldFilippov solutionHidden dynamicsPiecewise-smooth systemsRegularizationScaling invariance

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Area of Science:

  • Numerical analysis
  • Dynamical systems theory
  • Control theory

Background:

  • Piecewise-smooth dynamical systems present challenges in numerical treatment.
  • Classical solutions and sliding modes require robust numerical methods.
  • Non-uniqueness in solutions complicates analysis and simulation.

Purpose of the Study:

  • To develop a numerical algorithm for piecewise-smooth dynamical systems.
  • To address classical solutions and sliding modes up to codimension-2.
  • To provide a method for selecting unique solutions in cases of non-uniqueness.

Main Methods:

  • Development of a novel algorithm for numerical treatment.
  • Focus on selecting the formal limit solution of a regularized problem.
  • Avoidance of direct numerical solution of regularized differential equations.

Main Results:

  • The presented algorithm successfully handles piecewise-smooth dynamical systems.
  • It effectively selects a unique solution when multiple exist.
  • The method circumvents the stiffness and oscillations associated with regularized differential equations.

Conclusions:

  • The proposed algorithm offers an efficient and stable numerical treatment for piecewise-smooth dynamical systems.
  • It provides a practical approach to handling non-unique solutions.
  • This method simplifies the numerical simulation of complex dynamical systems.