Harnessing Adaptive Synchronization in Enzyme-Substrate Systems with Brain Wave-Like Ferroelectric Behavior

Abstract

This paper explores the application of chaos in various physical, electrical, chemical, and biological systems, with a specific focus on enzymes-substrate reactions exhibiting ferroelectric behaviour in brain waves. The study conducted by Enjieu Kadji, Chabi Orou, Yamapi, and Woafo (2007) is investi- gated, which examines the dynamic analysis and global chaos synchronization of a 2-D non-autonomous enzymes-substrates system subjected to a cosinusoidal forcing term. The phase por- traits of the chaotic behaviousr in the enzymes-substrates system are depicted. Furthermore, novel adaptive control techniques are developed to achieve global synchronization of identical enzyme-substrates systems with uncertain parameters. The main results for global synchronization are derived using backstepping control, and MATLAB plots are included to illustrate these findings.

Introduction

I. INTRODUCTION

The field of chaos theory focuses on studying the qualitative and numerical aspects of unstable and aperiodic behaviour in deterministic nonlinear dynamical systems. A system is considered chaotic if it exhibits boundedness, infinite recur- rence, and sensitive dependence on initial conditions. The pioneering work of Lorenz and Ro¨ssler in the 1960s and 1970s led to the discovery of well-known chaotic systems, and since then, numerous other chaotic systems have been identified. In the context of biological systems, Frohlich’s research on coherent oscillations and the presence of long- wavelength electric vibrations in active biological systems laid the groundwork for investigating enzymatic substrate reactions with ferroelectric behaviour in brain waves. Enjieu Kadji, Chabi Orou, Yamapi, and Woafo (2007) developed a model for enzymes-substrate reactions with ferroelectric behaviour in brain waves, specifically noting chaotic behaviour in the 2-D enzyme-substrate reactions system. This paper explores the chaotic properties of the enzyme-substrate reactions sys- tem, providing phase portraits of the chaotic system through MATLAB plots. The application of chaos and control theory extends to various scientific and engineering fields, including oscillators, memristors, biology, chemical reactions, circuits, and more. Recent research has focused on synchronizing chaotic systems, where a master-slave system approach is employed to ensure that the slave system asymptotically follows the trajectories of the master system. Numerous methods, such as active control, adaptive control, sliding mode control, and backstepping control, have been developed for chaotic system synchronization. In line with these developments, this paper introduces new results for the global chaos synchronization of enzymes-substrate systems using an adaptive backstepping controller design. The proposed approach leverages Lyapunov stability theory to establish the effectiveness of the controller. Enzymes-Substrates             Reaction System  TheEnzymes- Substrates Reaction System is a non-autonomous 2-D system with a cosinusoidal forcing term, which exhibits chaotic behaviour. This system models the behaviour of enzymes and substrates in biological systems. The dynamics of the system are described by a set of differential equations, where the concentrations of the enzymes and substrates change over time due to various reactions and interactions. The system is complex and nonlinear, and its behaviour is difficult to predict. However, the study of chaos in this system has important implications for understanding biological processes and developing new treatments for diseases. In particular, the synchronization of identical enzymes-substrates systems with uncertain parameters is of interest, and adaptive backstepping control techniques have been developed to achieve this goal. The Enzymes-Substrates Reaction System is an important model for studying the dynamics of biological systems and the application of chaos theory in science and engineering. The enzyme-substrate reactions system with ferroelectric behaviour in brain waves, derived by Enjieu Kadji, Chabi Orou, Yamapi, and Woafo, can be represented by the following differential equation:

Conclusion

This research paper introduces novel findings regarding the enzymes-substrates reaction system, specifically its ferroelec- tric behavior in brain waves, as discovered by Enjieu Kadji, Chabi Orou, Yamapi, and Woafo in 2007. The paper provides a comprehensive description and dynamic analysis of the sys- tem’s chaotic 2-D non-autonomous attractor. Furthermore, the study presents innovative outcomes concerning the adaptive chaos synchronization of identical enzymes-substrates reaction systems that involve uncertain parameters. The theoretical proofs are based on backstepping control and Lyapunov sta- bility theory. MATLAB simulations are employed to validate and illustrate the main findings.

References

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Copyright

Copyright © 2024 Amarpreet Kaur, Dr. Geeta Jain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.