Computing Lyapunov Exponents for Fractional-Order Differential Equations

This paper focuses on computing Lyapunov exponents for fractional-order differential equations and utilizing these exponents to identify chaotic states in dynamical systems. Chaotic systems represent a class of nonlinear dynamical systems exhibiting extreme sensitivity to initial conditions, making them significant in both theoretical research and practical applications. Our study investigates fundamental concepts and properties of chaotic systems, along with numerical methods for Lyapunov exponent calculation. The implementation typically involves solving fractional differential equations using Grünwald-Letnikov or Caputo derivative approximations, followed by Jacobian matrix computation for local linearization. Key algorithms include Wolf's method for exponent extraction from time series data and continuous orthogonalization techniques for maintaining numerical stability. Furthermore, we examine applications of chaotic systems in cryptography, secure communications, and image processing, where Lyapunov exponents serve as quantitative measures for chaos strength and system predictability. Through this research, we aim to deepen the understanding of chaotic system fundamentals and their potential applications.