一类随机耦合 Lorenz 映射的渐近行为

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 939-948.

一类随机耦合 Lorenz 映射的渐近行为

郑博文, 张亮^*()

武汉理工大学数学与统计学院武汉 430070

收稿日期:2025-02-26 修回日期:2025-06-19 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 张亮 E-mail:zhangl@whut.edu.cn
基金资助:
国家自然科学基金(11223344)

The Asymptotic Behavior of a Class of Randomly Coupled Lorenz Maps

Bowen Zheng, Liang Zhang^*()

Department of Mathematics, Wuhan University of Technology, Wuhan 430070

Received:2025-02-26 Revised:2025-06-19 Online:2026-06-26 Published:2026-06-16
Contact: Liang Zhang E-mail:zhangl@whut.edu.cn
Supported by:
NSFC(11223344)

摘要/Abstract

摘要：

该文研究一类分段线性 Lorenz 映射的混合性及其随机耦合映射的绝对连续不变测度存在性. 应用重整化理论, 该文严格证明了该 Lorenz 框架下两个确定性映射具有混合性并在此基础上设计数值实验发现特殊的同步混沌现象, 即两个确定性映射的 Lyapunov 指数均为正值, 但相应随机映射的 Lyapunov 指数为负值且仍存在绝对连续不变测度. 进一步研究表明, 该随机映射不仅具有全局遍历性, 还能诱导同步现象的产生.

关键词: Lorenz 映射, 平凡重整化, 随机映射, 绝对连续不变测度.

Abstract:

This paper investigates the mixing property of a class of piecewise linear Lorenz maps and the existence of absolutely continuous invariant measures for their stochastically coupled counterparts. Applying renormalization theory, it is rigorously proved that two deterministic maps within this Lorenz framework exhibit mixing. Based on this foundation, numerical experiments are designed, revealing a peculiar phenomenon of synchronized chaos: while both deterministic maps have positive Lyapunov exponents, the corresponding stochastic map possesses a negative Lyapunov exponent while still maintaining an absolutely continuous invariant measure. Further research demonstrates that this stochastic map not only possesses global ergodicity but can also induce the emergence of synchronization phenomena.

Key words: Lorenz map, trivial renormalization, random map, absolutely continuous invariant measure.

中图分类号:

O193

郑博文, 张亮. 一类随机耦合 Lorenz 映射的渐近行为[J]. 数学物理学报, 2026, 46(3): 939-948.

Bowen Zheng, Liang Zhang. The Asymptotic Behavior of a Class of Randomly Coupled Lorenz Maps[J]. Acta mathematica scientia,Series A, 2026, 46(3): 939-948.

图/表 5

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