Spectral decomposition of nonlinear systems with memory.

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.