Liapunov Exponents and Control Theory-Based Stability Analysis and Parameter Optimization Technique for Dynamical Systems with Periodic Variable Coefficients

INTRODUCTION: This is the introductory text Dynamical systems with periodic variable coefficients find extensive use in physics and engineering domains, including nonlinear circuits, structural dynamics, and vibration control. The intricacy of the parameter changes over time has made the stability and control issues of such systems a popular topic in engineering and academics. The dynamic features of periodic variable coefficient dynamical systems are difficult to adequately characterize using approaches based on standard stability theory, as demonstrated by previous research. This work formulates a methodology that integrates Lyapunov exponents, Floquet theory, and analysis of fractional-order systems to evaluate the stability of variable-coefficient periodic systems. The effectiveness of the approach is illustrated in nonlinear control and bifurcation analysis. OBJECTIVES: This is particularly true when the parameters vary significantly or the system behaves nonlinearly. As crucial instruments for examining dynamical systems, Lyapunov stability theory and Floquet theory are essential for determining global stability and studying periodic systems, respectively. Further research is still required to determine the best way to integrate the two theories in order to provide straightforward and useful conclusions for periodic systems with variable coefficients. METHODS: In the meantime, fractional-order systems have drawn interest recently in the fields of control and chaotic dynamics due to their precise representation of genetic characteristics and memory effects. It has been demonstrated that under parameter changes, fractional-order chaotic systems display a variety of dynamic behaviors, such as multistability, bifurcation phenomena, and complexity shifts. By logically creating control techniques, such systems can be optimized to increase their robustness and stability while also exposing the complex system's dynamical principles. RESULTS: In light of this, this research suggests an integrated framework for a thorough investigation of the stability of dynamical systems with periodic variable coefficients that combines the Lyapunov eigenindex, Lyapunov function, and fractional order complexity analysis. In particular, this work first builds a stability criterion to theoretically support periodic systems using Floquet theory with Lyapunov exponents; The practical issues of modular multilevel dc voltage regulators (MMC-DVR) are then addressed by a nonlinear control method based on Lyapunov functions. Additionally, bifurcation diagrams with complexity index (such as spectral entropy complexity) are used for fractional-order chaotic systems in order to examine the impact of parameter changes on system stability and chaotic behavior. Lastly, numerical examples are used to confirm the efficacy of the suggested approach. CONCLUSION: It is demonstrated that the analytical framework put forth in this research may successfully address challenging issues in the control and stability design of dynamical systems with periodic variable coefficients while also offering fresh concepts for the optimization of intricate system parameters.