分数阶自治动力学系统初值问题的递推公式及其应用

doi:10.11772/j.issn.1001-9081.2024030289

《计算机应用》唯一官方网站 ›› 2025, Vol. 45 ›› Issue (2): 556-562.DOI: 10.11772/j.issn.1001-9081.2024030289

• 先进计算 • 上一篇

分数阶自治动力学系统初值问题的递推公式及其应用

符五久, 周林, 邓建杰, 游泳()

北京理工大学（珠海）数理与土木工程学院，广东珠海 519088

收稿日期:2024-03-20 修回日期:2024-04-23 接受日期:2024-04-24 发布日期:2024-05-21 出版日期:2025-02-10
通讯作者: 游泳
作者简介:符五久（1956—），男，安徽无为人，教授，主要研究方向：混沌动力学
周林（1979—），男，黑龙江哈尔滨人，副教授，博士，主要研究方向：凝聚态物理、混沌动力学
邓建杰（1982—），男，湖南平江人，副教授，硕士，主要研究方向：非线性动力学；

Recurrence formula for initial value problems of fractional-order autonomous dynamics system and its application

Wujiu FU, Lin ZHOU, Jianjie DENG, Yong YOU()

School of Applied Science and Civil Engineering，Beijing Institute of Technology，Zhuhai，Zhuhai Guangdong 519088，China

Received:2024-03-20 Revised:2024-04-23 Accepted:2024-04-24 Online:2024-05-21 Published:2025-02-10
Contact: Yong YOU
About author:FU Wujiu， born in 1956， professor. His research interests include chaotic dynamics.
ZHOU Lin， born in 1979， Ph. D.， associate professor. His research interests include condensed matter physics， chaotic dynamics.
DENG Jianjie， born in 1982， M. S.， associate professor. His research interests include non-linear dynamics.

摘要/Abstract

摘要：

对分数阶微分动力学系统进行数值计算时，直接离散微分方程存在长时程储存困难。为解决这一问题，首先，将微分方程做一次积分，然后再离散化；同时，给出一个递推公式，并讨论它的适用条件。用该递推公式计算一些常见的非线性问题所得的结果都与其他数值方法的结果一致。由于二维分数阶连续动力学系统是否有混沌运动尚未有定论，应用这个递推公式对二维连续耦合Logistic模型进行研究，发现该系统仅由平衡点通过Hopf分岔产生极限环，不存在混沌运动。最后，给出分数阶二维连续Logistic系统运动的李雅普诺夫指数判据。

关键词: 分数阶, 自治动力学系统, 初值问题, 递推公式, 二维连续Logistic系统

Abstract:

When calculating fractional-order differential dynamics systems numerically， there are difficulties of long-time memory storage by discretizing differential equation directly. In order to solve this problem， firstly， the differential equation was integrated once and then discretized. At the same time， a recurrence formula was given and its applicable conditions were discussed. Some common non-linear problems were calculated by this formula. The results of the above were consistent with those of other numerical methods. As whether there is chaotic motion in two-dimensional fraction-order continuous dynamics system is not concluded， this recurrence formula was used to study the two-dimensional continuous coupled Logistic model. It is found that there is only the limit cycle generated by the equilibrium point through Hopf bifurcation in this system without chaotic motion. Finally， the Lyapunov exponent criterion for the motion of two-dimensional fractional-order continuous Logistic system was given.

Key words: fractional-order, autonomous dynamics system, initial value problem, recurrence formula, two-dimensional continuous Logistic system

中图分类号:

TP391.9

符五久, 周林, 邓建杰, 游泳. 分数阶自治动力学系统初值问题的递推公式及其应用[J]. 计算机应用, 2025, 45(2): 556-562.

Wujiu FU, Lin ZHOU, Jianjie DENG, Yong YOU. Recurrence formula for initial value problems of fractional-order autonomous dynamics system and its application[J]. Journal of Computer Applications, 2025, 45(2): 556-562.

图/表 13

图1 Logistic方程的解曲线

Fig. 1 Solution curve of Logistic equation

图2 时滞Logistic方程的时间序列曲线

Fig.2 Time series curve of time-delay Logistic equation

图3 分数阶系统（19）的解曲线

Fig. 3 Solution curves of fractional-order system （19）

图4 分数阶时滞系统（21）的解曲线

Fig. 4 Solution curves of fractional-order time-delay system （21）

图5 解曲线随α的变化

Fig.5 Change of solution curves varying with parameter α

图6 解曲线随r变化

Fig. 6 Change of solution curves varying with parameter r

图7 含时滞的两种群Lotka-Volterra系统的相轨迹

Fig. 7 Phase trajectory of two-population Lotka-Volterra system with time delay

图8 整数阶Lorenz混沌吸引子

Fig. 8 Integer-order Lorenz chaotic attractors

图9 分数阶Lorenz混沌吸引子

Fig. 9 Fractional-order Lorenz chaotic attractors

图10 二维耦合Logistic系统（30）的分岔图

Fig. 10 Bifurcation diagrams of two-dimensional coupled Logistic system （30）

图11 二维耦合Logistic系统（30）的李雅普诺夫指数谱

Fig. 11 Lyapunov exponent spectrum of two-dimensional coupled Logistic system （30）

图12 二维耦合分数阶Logistic系统（30）的李雅普诺夫指数谱

Fig. 12 Lyapunov exponent spectrum of two-dimensional coupled fractional-order Logistic system （30）

图13 二维耦合分数阶Logistic系统（30）的分岔图

Fig. 13 Bifurcation diagram of two-dimensional coupled fractional Logistic system （30）

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分数阶自治动力学系统初值问题的递推公式及其应用

Recurrence formula for initial value problems of fractional-order autonomous dynamics system and its application

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