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Spectral decomposition of nonlinear systems with memory.

Adam Svenkeson1, Bryan Glaz1, Samuel Stanton2

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This study introduces a new method for analyzing nonlinear systems with long-term memory using Koopman operator and Lévy transformation. This approach reveals anomalous temporal behavior and memory effects in dynamical systems.

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Area of Science:

  • Dynamical Systems Analysis
  • Nonlinear System Theory
  • Fractional Calculus

Background:

  • Traditional spectral methods struggle with nonlinear systems exhibiting long-term memory.
  • Memory effects in dynamical systems are often attributed to environmental interactions.
  • Understanding these memory effects is crucial for accurate system modeling.

Purpose of the Study:

  • To develop an alternative analytical approach for nonlinear systems with long-term memory.
  • To decompose memory-laden systems into modes with anomalous temporal dynamics.
  • To demonstrate the utility of this method for analyzing ill-defined (black-box) systems.

Main Methods:

  • Utilizing the Koopman operator framework.
  • Applying a Lévy transformation in time.
  • Employing fractional calculus for system description.
  • Spectral decomposition using Mittag-Leffler functions.

Main Results:

  • Decomposition of nonlinear systems into modes with anomalous, scale-free temporal behavior.
  • Average mode evolution follows a Mittag-Leffler function.
  • Demonstrated applicability on fractional harmonic oscillator and logistic equations.
  • Identification of hidden memory effects in black-box systems via spectral analysis.

Conclusions:

  • The proposed method effectively uncovers inherent memory effects in dynamical systems.
  • Mittag-Leffler function-based spectral decomposition offers a powerful tool for analyzing complex systems.
  • This approach aids in determining the necessity of memory operators in numerical modeling when system details are unknown.