arXiv 2408.163522024-08-31nlin.CD

在简单离散时间忆阻器电路中嵌入经典混沌映射

Embedding classic chaotic maps in simple discrete-time memristor circuits

Mauro Di Marco, Mauro Forti, Giacomo Innocenti, Luca Pancioni, Alberto Tesi

AI总结：

本文利用DT-FCAM证明最简单的离散时间忆阻器电路可在不变流形上精确嵌入logistic、帐篷、Hénon和Lozi等经典混沌映射，从而解释了简单DT忆阻器电路呈现复杂混沌动力学的原因。

摘要：

过去几年，文献中对由离散时间（DT）忆阻器电路实现的混沌映射的兴趣出现了显著激增。本文探究了这类混沌行为背后的原因。为此，本文考虑了由电容器和理想磁通控制忆阻器、或电感器和理想电荷控制忆阻器构成的最简单忆阻器电路所实现的映射。特别地，本文使用最近一篇论文中引入的DT磁通-电荷分析方法（FCAM），以确保连续时间（CT）忆阻器电路的第一积分和不变流形上的叶层结构在任意步长离散化中都能被精确保持。DT-FCAM在电压-电流域（VCD）中产生一个二维映射，并在磁通-电荷域（FCD）中产生一个依赖于流形的一维映射，即在每个不变流形上的一维映射。一个主要结果是，在适当选择电路参数和忆阻器非线性特性的情况下，两种DT电路都能精确嵌入两个经典混沌映射，即logistic映射和帐篷映射。此外，由于极端多稳态特性，DT电路能够在流形中同时嵌入通过改变logistic映射和帐篷映射中一个参数所展现的全部动力学行为。本文随后考虑了一个DT忆阻器Murali-Lakshmanan-Chua电路及其对偶电路。通过DT-FCAM，这些电路在VCD中实现一个三维映射，并在FCD中的每个不变流形上实现一个二维映射。研究表明，当改变Hénon映射和Lozi映射中的一个参数时，两种电路都能在流形中同时嵌入这两个经典混沌映射所展现的全部动力学行为。本质上，这些结果解释了为什么即使在简单的DT忆阻器电路中观察到复杂动力学行为也不令人惊讶。

英文摘要：

In the last few years the literature has witnessed a remarkable surge of interest for chaotic maps implemented by discrete-time (DT) memristor circuits. This paper investigates on the reasons underlying this type of chaotic behavior. To this end, the papers considers the map implemented by the simplest memristor circuit given by a capacitor and an ideal flux-controlled memristor or an inductor and an ideal charge-controlled memristor. In particular, the manuscript uses the DT flux-charge analysis method (FCAM) introduced in a recent paper to ensure that the first integrals and foliation in invariant manifolds of continuous-time (CT) memristor circuits are preserved exactly in the discretization for any step size. DT-FCAM yields a two-dimensional map in the voltage-current domain (VCD) and a manifold-dependent one-dimensional map in the flux-charge domain (FCD), i.e., a one-dimensional map on each invariant manifold. One main result is that, for suitable choices of the circuit parameters and memristor nonlinearities, both DT circuits can exactly embed two classic chaotic maps, i.e., the logistic map and the tent map. Moreover, due to the property of extreme multistability, the DT circuits can simultaneously embed in the manifolds all the dynamics displayed by varying one parameter in the logistic and tent map. The paper then considers a DT memristor Murali-Lakshmanan-Chua circuit and its dual. Via DT-FCAM these circuits implement a three-dimensional map in the VCD and a two-dimensional map on each invariant manifold in the FCD. It is shown that both circuits can simultaneously embed in the manifolds all the dynamics displayed by two other classic chaotic maps, i.e., the Henon map and the Lozi map, when varying one parameter in such maps. In essence, these results provide an explanation of why it is not surprising to observe complex dynamics even in simple DT memristor circuits.

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摘要：

英文摘要：

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