Discretization process of coupled Logistic fractional-order differential equation

doi:10.11772/j.issn.1001-9081.2018040848

Abstract

Abstract: Focusing on the problem of solving coupled Logistic fractional-order differential equation, a discretization method was introduced to solve it discretly. Firstly, a coupled Logistic integer-order differential equation was introduced into the fields of fractional-order calculus. Secondly, the corresponding coupled Logistic fractional-order differential equation with piecewise constant arguments was analyzed and the proposed discretization method was applied to solve the model numerically. Then, according to the fixed point theory, the stability of the fixed point of the synthetic dynamic system was discussed, and the boundary equation of the first bifurcation of the coupled Logistic fractional-order system in the parameter space was given. Finally, the model was numerically simulated by Matlab, and more complex dynamics phenomena of model were discussed with Lyapunov index, phase diagram, time series diagram and bifurcation diagram. The simulation results show that, the proposed method is successful in discretizing coupled Logistic fractional-order differential equation.

Key words: coupled Logistic differential equation, time delay, piecewise constant argument, fixed point, bifurcation, chaos

摘要： 针对二维Logistic分数阶微分方程的求解问题，引进了一种离散化方法对其进行离散求解。首先，将二维Logistic整数阶微分方程推广到分数阶微积分领域；其次，分析相应具有分段常数变元的二维Logistic分数阶微分方程并应用提出的离散化方法对模型进行数值求解；然后，根据不动点理论讨论该合成动力系统不动点的稳定性，给出了在参数空间内二维Logistic分数阶系统发生第一次分岔的边界方程；最后，借助Matlab对模型进行数值仿真，并结合Lyapunov指数、相图、时间序列图、分岔图探讨模型更多复杂的动力学现象。仿真结果显示，所提方法成功对二维Logistic分数阶微分方程进行离散。

关键词: 二维Logistic微分方程, 时滞, 分段常数变元, 不动点, 分岔, 混沌

CLC Number:

LIU Shanshan, GAO Fei, LI Wenqin. Discretization process of coupled Logistic fractional-order differential equation[J]. Journal of Computer Applications, 2019, 39(1): 305-310.

刘杉杉, 高飞, 李文琴. 二维Logistic分数阶微分方程的离散化过程[J]. 计算机应用, 2019, 39(1): 305-310.

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