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Reducing nonlinear dynamical systems to canonical forms.

Léon Brenig1

  • 1Service de Physique des Systèmes Dynamiques, Faculté des Sciences, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium lbrenig@ulb.ac.be.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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Summary
This summary is machine-generated.

A new framework transforms nonlinear dynamical systems into canonical forms, enabling analysis of integrability, stability, and solutions. This approach links nonlinear dynamics to stochastic processes via abstract Lie algebra.

Keywords:
Lie algebraLotka–VolterraTaylor seriescanonical formnonlinear dynamicsurn process

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Area of Science:

  • Mathematics
  • Physics
  • Dynamical Systems Theory

Background:

  • Nonlinear differential dynamical systems are prevalent in science.
  • Existing methods for analyzing these systems can be limited.
  • A unified approach is needed for broader applicability.

Purpose of the Study:

  • To present a global framework for treating nonlinear differential dynamical systems.
  • To establish canonical forms for analyzing system properties.
  • To connect nonlinear dynamics with stochastic processes.

Main Methods:

  • Transformation of systems into a quasi-polynomial format.
  • Characterization using Lotka-Volterra (LV) and monomial canonical forms.
  • Application of abstract Lie algebra and its realizations (e.g., bosonic operators).

Main Results:

  • Identified infinite equivalence classes for systems in quasi-polynomial format.
  • Developed methods for finding integrability conditions, invariants, and dimension reductions.
  • Derived Lyapunov functions and stability analysis tools from the LV form.
  • Obtained analytic forms for Taylor series solutions, generating new special functions.
  • Proved an equivalence between urn processes and dynamical systems.

Conclusions:

  • The proposed framework offers a unified approach to nonlinear dynamical systems.
  • The framework reveals underlying abstract Lie algebra structures.
  • New connections are established between nonlinear dynamics, stochastic processes, and special functions.