Research on the Lowest Cost to Calculate the Lyapunov Exponents from Fractional Differential Equations

Background Introduction

Fractional Differential Equations (FDEs) extend traditional calculus by allowing derivatives and integrals of non-integer orders. This mathematical framework exhibits unique advantages in describing complex dynamical behaviors, particularly in the study of chaotic and nonlinear systems. Lyapunov Exponents (LEs) are critical indicators for measuring a system’s sensitivity to initial conditions and are commonly used to determine whether a system is in a chaotic state. However, calculating Lyapunov exponents for fractional-order chaotic systems is often computationally expensive, especially in high-dimensional systems. Therefore, reducing computational costs and improving efficiency has become a significant challenge in the study of fractional-order chaotic systems.

This paper, authored by Shuang Zhou, Qiyin Zhang, Shaobo He, and Yingqian Zhang, aims to systematically investigate the computational cost of Lyapunov exponents in fractional differential equations using the Adomian Decomposition Method (ADM). By analyzing the effects of the number of decomposition terms and iteration step sizes on the calculation of Lyapunov exponents, the study proposes strategies to optimize computational efficiency and accuracy.

Source of the Paper

  • Authors: Shuang Zhou, Qiyin Zhang, Shaobo He, Yingqian Zhang
  • Institutions: Shuang Zhou and Qiyin Zhang are affiliated with Guizhou University and Chongqing Normal University, China, respectively; Shaobo He is from Xiangtan University; and Yingqian Zhang is from Xiamen University Malaysia.
  • Submission and Acceptance Dates: Submitted on September 30, 2024, and accepted on January 27, 2025.
  • Journal: Published in a journal under Springer Nature B.V.

Research Process

1. Research Objectives and Methods

The study aims to optimize the calculation of Lyapunov exponents in fractional differential equations using the Adomian Decomposition Method. It focuses on the impact of the number of decomposition terms and iteration step sizes on computational results. By analyzing three-dimensional and four-dimensional fractional chaotic systems, the research proposes optimal numbers of decomposition terms and iteration step sizes to improve computational efficiency and accuracy.

2. Experimental Design and Steps

The study selects classic systems as research objects, including the three-dimensional Chen system, the simplified Lorenz system, the modified Van der Pol-Duffing system (MADVP system), and the Hastings-Powell food chain model (HP model). The specific steps are as follows:

a) System Modeling and Decomposition

First, the fractional differential equations of each system are decomposed using the Adomian Decomposition Method. The ADM expands nonlinear terms into polynomials, representing the system’s solution as an infinite series. The study focuses on the impact of the number of decomposition terms, ranging from 2 to 7 terms.

b) Iterative Calculation and Lyapunov Exponent Calculation

For each system, the study calculates Lyapunov exponents using the QR decomposition method. By adjusting iteration step sizes (h = 0.01, 0.001, 0.0001), the study analyzes the computational results of Lyapunov exponents under different step sizes. Additionally, the study computes the system’s Jacobian matrix using MATLAB software and calculates Lyapunov exponents based on the QR decomposition method.

c) Analysis of Chaotic Attractor Trajectories

The study also examines the changes in chaotic attractor trajectories under different numbers of decomposition terms and iteration step sizes. By plotting attractor trajectory diagrams, the research validates the impact of the number of decomposition terms and iteration step sizes on the system’s dynamical behavior.

3. Key Findings

a) Impact of the Number of Decomposition Terms

The results show that the number of decomposition terms significantly affects the calculation of Lyapunov exponents. In the three-dimensional Chen system and the simplified Lorenz system, the most accurate results for Lyapunov exponents are obtained when the number of decomposition terms is 3. When the number of terms is reduced to 2, the Lyapunov exponents exhibit significant deviations and fail to accurately capture chaotic behavior.

b) Impact of Iteration Step Sizes

Iteration step sizes also play a crucial role in the accuracy of Lyapunov exponent calculations. Smaller step sizes (e.g., h = 0.001 or 0.0001) significantly improve numerical stability, especially when the number of decomposition terms is low. The study also finds that larger step sizes (e.g., h = 0.01) may lead to non-chaotic states, making it impossible to calculate Lyapunov exponents.

c) Chaotic Attractor Trajectories

By plotting chaotic attractor trajectories, the study validates the impact of the number of decomposition terms and iteration step sizes on the system’s dynamical behavior. When the number of decomposition terms is 3 and the step size is small, the chaotic attractor trajectories closely match those obtained with a higher number of terms.

4. Conclusions and Significance

This study systematically analyzes the computational cost of Lyapunov exponents in fractional differential equations using the Adomian Decomposition Method. It proposes that in three-dimensional and four-dimensional fractional chaotic systems, using 3 decomposition terms and smaller step sizes can significantly improve computational efficiency and accuracy. These findings provide important optimization strategies for the numerical calculation of fractional chaotic systems, offering high scientific and practical value.

Research Highlights

  1. Optimization of the Number of Decomposition Terms: The study is the first to propose that using 3 decomposition terms in fractional chaotic systems strikes a balance between computational efficiency and accuracy, providing a valuable reference for future research.
  2. Impact of Iteration Step Sizes: The study finds that smaller iteration step sizes significantly enhance numerical stability, particularly when the number of decomposition terms is low.
  3. Broad Applicability: The results are not only applicable to classic three-dimensional and four-dimensional chaotic systems but can also be extended to higher-dimensional systems, demonstrating high universality.

Other Valuable Information

The study also reveals that the choice of the number of decomposition terms and iteration step sizes significantly affects the system’s lowest order. By optimizing these parameters, the research enables more accurate determination of the system’s lowest order, providing new insights for the modeling and analysis of fractional chaotic systems.

This study systematically optimizes the calculation of Lyapunov exponents in fractional differential equations using the Adomian Decomposition Method, offering important theoretical and methodological support for research on fractional chaotic systems.