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Singular bifurcations and regularization theory.

Alexander Farutin1, Chaouqi Misbah1

  • 1<a href="https://ror.org/02rx3b187">Université Grenoble Alpes</a>, CNRS, <a href="https://ror.org/023n9q531">LIPhy</a>, F-38000 Grenoble, France.

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Summary
This summary is machine-generated.

This study introduces singular bifurcations, a new concept in nonlinear science, challenging traditional analysis methods. It presents a universal theory to handle these complex bifurcations, expanding our understanding of nonlinear systems.

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Area of Science:

  • Nonlinear Sciences
  • Mathematical Physics
  • Active Matter Physics

Background:

  • Nonlinear science is crucial across diverse disciplines, from physics to social sciences.
  • Bifurcation analysis traditionally relies on regular perturbative expansions.
  • Hidden singularities in models challenge the universality of regular expansions.

Purpose of the Study:

  • To introduce and define the concept of singular bifurcations in nonlinear science.
  • To develop a universal theory for handling and regularizing singular bifurcations.
  • To illustrate the theory with an example from active matter systems.

Main Methods:

  • Analysis of systems exhibiting hidden singularities near bifurcation points.
  • Development of a regularization theory for singular bifurcations.
  • Application of the theory to phoretic microswimmer models.

Main Results:

  • Demonstration of singular bifurcations in active matter systems.
  • Establishment of a universal framework for analyzing and regularizing these bifurcations.
  • Identification of a previously overlooked aspect of nonlinear science.

Conclusions:

  • Regular perturbative expansions are not universally applicable for bifurcation analysis.
  • Singular bifurcations represent a significant and overlooked phenomenon in nonlinear dynamics.
  • The proposed universal theory provides a novel approach to understanding complex nonlinear systems.