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Stability analysis of fractional difference equations with delay.

Divya D Joshi¹, Sachin Bhalekar², Prashant M Gade¹

¹Department of Physics, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur 440033, India.

Chaos (Woodbury, N.Y.)

|May 8, 2024

Summary

This study analyzes stability conditions for fractional difference equations with time delays, crucial for modeling long-term memory in various systems. The findings extend to nonlinear systems and offer a framework for complex delayed dynamics.

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Area of Science:

Dynamical Systems and Control Theory
Mathematical Modeling
Computational Neuroscience

Background:

Long-term memory is a key feature in diverse systems, often modeled using time delays.
Fractional order differences introduce nonlocal behavior, contributing to long-time memory effects.
Fractional difference equations with delay are suitable for modeling systems exhibiting memory, yet their stability analysis is underexplored.

Purpose of the Study:

To derive and analyze stability conditions for linear fractional difference equations with arbitrary and distributed time delays.
To extend the stability analysis to nonlinear fractional difference systems.
To provide a foundational framework for understanding the dynamics of fractional order systems with memory.

Main Methods:

Derivation of stability conditions for linear fractional difference equations with arbitrary delay (τ).
Analysis of systems with distributed delays.
Detailed stability analysis for specific single delay cases (τ=1, τ=2).
Extension of derived conditions to nonlinear fractional difference maps.

Main Results:

Established stability conditions for fractional difference equations with arbitrary and distributed delays.
Provided detailed stability analysis for single delays τ=1 and τ=2.
Demonstrated the applicability of the derived formalism to nonlinear systems.
Showcased the extensibility of the framework to multiple time delays.

Conclusions:

The study provides crucial stability criteria for fractional difference equations with delays, enhancing the modeling of memory in complex systems.
The developed methods are applicable to both linear and nonlinear fractional order systems with various delay configurations.
This work lays the groundwork for further research into the stability and dynamics of fractional systems with memory.