Dimensionality Reduction Analysis for Detecting Initial Effects on Synchronization of Memristor-Coupled System

Yunzhen Zhang,Bocheng Bao,Huagan Wu,Shengyao Chen,Zhong Liu

doi:10.1109/access.2019.2933252

Abstract

This paper presents a 9-D memristor-coupled system with three ideal memristors and investigates its initial effects on synchronization using dimensionality reduction analysis method. The 9-D memristor-coupled system is yielded from two identical 4-D ideal memristor-based hyper-jerk systems via coupling an ideal memristor, from which the initials-dependent synchronization with parallel offset for larger coupling strength is numerically exhibited. To explore the initial effects on synchronization, an equivalent 6-D dimensionality reduction model is built using state variable mapping (SVM) method, from which the implicit initials of the 9-D memristor-coupled system are transformed into the explicitly initials-related system parameters of such a 6-D dimensionality reduction model. Thus, the inherent initial mismatches between the two coupled identical 4-D subsystems are explicitly expressed as the initials-related parameter mismatches between the two coupled non-identical 3-D dimensionality reduction subsystems. The initials-related boundedness of the error system between the two non-identical 3-D dimensionality reduction subsystems is derived by Lyapunov analysis method, upon which the initial effects on synchronization with parallel offset are expounded quantitatively. Furthermore, the initials-dependent synchronization is well confirmed by the numerical simulations, which demonstrates that the initials do have great influence on synchronization dynamics of the coupled memristive system.

Highlights

Due to the nature nonlinearity [1], [2], memristors were usually introduced into some existing dynamical circuits and systems to construct different kinds of memristive dynamical circuits and systems, such as memristive HindmarshRose neuron model [3], memristive cellular nonlinear/neural network [4], memristive band-pass filter circuit [5], memristive jerk circuit [6], [7], memristive Twin-T oscillator [8], memristive multi-scroll Chua’s circuit [9], memristive logic circuit [10], memristive non-autonomous chaotic circuit [11], and so on
The inherent initial mismatches between the two coupled identical 4-D subsystems are formulated as the initials-related parameter mismatches between the two coupled non-identical 3-D dimensionality reduction subsystems
What needs illustration is that, under the situation Xi(0) = Yi(0) = V (0) = 0, system (8) exhibits the completely same dynamical behaviors as the presented system (2) [33]. It follows that the aforementioned 6-D dimensionality reduction model (8) can be utilized for quantitatively analyzing the initial effects on synchronization with parallel offset in system (2) by changing the initials-related system parameters δi, ηi and v0

Summary

INTRODUCTION

Due to the nature nonlinearity [1], [2], memristors were usually introduced into some existing dynamical circuits and systems to construct different kinds of memristive dynamical circuits and systems, such as memristive HindmarshRose neuron model [3], memristive cellular nonlinear/neural network [4], memristive band-pass filter circuit [5], memristive jerk circuit [6], [7], memristive Twin-T oscillator [8], memristive multi-scroll Chua’s circuit [9], memristive logic circuit [10], memristive non-autonomous chaotic circuit [11], and so on. The initials-related boundedness of the error system between the two non-identical 3-D dimensionality reduction subsystems is derived theoretically by Lyapunov analysis method, upon which the initial effects on synchronization with parallel offset are thereby quantitatively explored. B. INITIALS-DEPENDENT SYNCHRONIZATION WITH PARALLEL OFFSET Complete synchronization occurs when the two coupled systems asymptotically exhibit identical dynamical behaviors, i.e., ||xi(t) − yi(t)|| → 0 as t → ∞. What needs illustration is that, under the situation Xi(0) = Yi(0) = V (0) = 0, system (8) exhibits the completely same dynamical behaviors as the presented system (2) [33] It follows that the aforementioned 6-D dimensionality reduction model (8) can be utilized for quantitatively analyzing the initial effects on synchronization with parallel offset in system (2) by changing the initials-related system parameters δi, ηi and v0. The solution of system (10) is uniformly bounded for all t ≥ t0 and uniformly bounded with the ultimate bound h−111(h12(η))

QUANTITATIVE ANALYSIS FOR INITIAL EFFECTS ON

CONCLUSION