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Reversible chemical reactions model with fractional difference operator: Dynamical analysis and synchronization.

D Vignesh¹, Santo Banerjee²

¹Cyber Security and Digital Industrial Revolution Centre, Universiti Pertahanan Nasional Malaysia, Kem, Sungai Besi, 57000 Kuala Lumpur, Malaysia.

Chaos (Woodbury, N.Y.)

|April 1, 2023

Summary

This study models two-step reversible chemical reactions using fractional calculus, revealing chaotic dynamics and enabling subsystem synchronization. These findings have applications in biofuel production from vegetable oils and animal fats.

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Area of Science:

Chemical kinetics
Mathematical modeling
Fractional calculus

Background:

Chemical reactions are fundamental to life, with understanding their complex dynamics crucial for real-world applications.
Reversible chemical reactions, like ester hydrolysis and salt formation, are vital in industrial processes.

Purpose of the Study:

To develop a mathematical model for two-step reversible chemical reactions using a Caputo fractional difference operator.
To analyze the chaotic behavior and synchronization of the proposed chemical reaction model.

Main Methods:

Construction of a mathematical model incorporating a Caputo fractional difference operator.
Analysis of chaotic responses using bifurcation diagrams and time-varying plots.
Investigation of system periodicity using maximum Lyapunov exponents and Jacobian matrix.
Achieving subsystem synchronization via nonlinear control functions.

Main Results:

Demonstrated chaotic behavior in the system for both identical and non-identical fractional orders.
Illustrated periodic state changes and confirmed chaos with Lyapunov exponents.
Successfully synchronized subsystems using nonlinear control strategies.
Presented numerical simulations comparing commensurate and incommensurate order models.

Conclusions:

The fractional-order mathematical model effectively captures the complex dynamics of two-step reversible chemical reactions.
The study provides insights into chaotic dynamics and synchronization, with potential applications in biofuel production.
Understanding these fractional-order dynamics is significant for optimizing transesterification reactions in biofuel synthesis.