Complex dynamics of a bi-directional N-type locally-active memristor

Highlights

• A novel bi-directional N-type locally-active memristor model is built.

• This memristor shows capacitive, inductive and resistive properties with different operating voltages via small-signal method.

• The memristor-based second-order periodic and third-order chaotic oscillators are designed and realized.

• The complexity theorem, no-complexity theorem and edge of chaos theorem are verified.

Abstract

This paper presents a bi-directional N-type locally-active memristor (LAM) model, which has two symmetrical locally-active regions with respect to the origin. For this memristor, the locally-active regions coincide with the edge of chaos regimes, where complex dynamic behaviors may occur. By deriving the small-signal admittance of the memristor, it is found that the LAM may be pure resistive, inductive or capacitive in terms of biasing voltages. When the LAM operates at the edge of chaos and connects with an external inductor in series, a second-order periodic oscillator is established. In this case, it is possible to move the poles of the system from the left-half plane to the right-half plane, thereby generating periodic oscillations. Complex dynamics of the second-order memristor-based oscillator is analyzed via Hopf bifurcation analysis and based on the theories of local activity and edge of chaos. Using the parasitic capacitor of the LAM in the second-order oscillator, a third-order chaotic oscillator is designed, which has a negative real pole and a pair of complex conjugate poles with positive real parts, satisfying the Shilnikov criterion. It is found that both periodic oscillation and chaos appear only at the edge of chaos of the LAM, thus verifying the complexity theorem, no-complexity theorem and edge of chaos theorem. Finally, circuit experiments are performed, which verify the dynamics and the effectiveness of the LAM model, the periodic oscillator and the chaotic oscillator.

Introduction

Local activity is the origin of complexity [1], and complex nonlinear dynamics may only emerge in locally-active systems [2]. In fact, local activity is necessary to supply signal energy amplification [3]. Besides, the action potential (spikes) generated from neurons in a neural network emerges near the edge of chaos [4], a subset of a locally-active region. Memristor, moreover, was coined by Chua in 1971 [5], as the fourth basic circuit element in addition to resistor, capacitor and inductor. In recent years, many research results have been reported, showing that the locally-active memristor (LAM) can be used to mimic a neuron possessing complex neuromorphic properties, such as periodic spiking, bursting, self-sustained oscillations, chaos and so on [6], [7].

In 2008, the Hewlett-Packard laboratory fabricated a nanoscale memristor [8], which stimulated unprecedented worldwide research interest because of its potential technological applications, such as chaotic circuits [9], [10], non-volatile memories [11], memristive neural networks [12], [13], and digital logic operations [14]. Nevertheless, the discovery of memristor with its laboratory prototype still face with technical challenges in physical realization and industrial production today.

A locally-active memristor is defined as any memristor that shows negative differential memductance or memristance in its DC V–I plot [15]. LAMs are classified into two types: current-controlled LAM and voltage-controlled LAM, which are called S-type LAM [16] and N-type LAM [17], respectively. Recently, some real S-type LAMs composing of different materials have been made [6], [7], [18]. Also, it has been shown that by incorporating the niobium dioxide (NbO₂) Mott LAM-based chaotic oscillator into a Hopfield neural network can greatly improve the converging efficiency and accuracy to yield a solution for computationally difficult problems [18]. In 2020, it is reported in [7] that an isolated third-order nanocircuit element, which can produce neuromorphic action-potential behaviors, paves a way towards densely functional neuromorphic computing primitives. Moreover, a two-channel vanadium dioxide (VO₂) LAM is constructed as a neuron, which can well emulate the neuronal dynamics [6].

In order to explore the characteristics of LAMs, Chua proposed a LAM, named Chua Corsage LAM, which is globally passive but locally active [19] and can generate periodic oscillations when connected with a passive inductor in series [20], [21]. Also, a second-order LAM is presented in [22], which generates periodic oscillations when connected in series with a battery only. These two LAM-based circuits, however, cannot give rise to chaos. In [23], a volatile LAM model is designed, which can generate chaos when connected in series with a capacitor and an inductor. Later, in [24], a non-volatile LAM is reported, which can generate chaos when connected with a capacitor and an inductor in parallel. These two LAMs are not globally passive, however, thereby it is impossible to be implemented without internal source.

Local-activity theorem provides a method to analyze the complexity of nonlinear circuits [25], [26]. It shows that local activity is the origin of complexity, which predicts the domain of complexity, namely, locally-active region where the edge of chaos is located. In [27], [28], a phase transition between order and chaos was revealed, where the transition boundary is referred to as the edge of chaos. In this region, the cells or systems have the greatest potential to support information storage and transmission, thereby realizing optimal computation [29], [30], [31]. Both local activity and edge of chaos theorems are essential to gain insight into the mechanism of chaos.

This paper proposes new methods for designing N-type LAM-based periodic and chaotic oscillators, and show why an LAM-based circuit could generate complex oscillations. First, an LAM model is proposed, which is passive but locally active. Then, a simplest periodic oscillator is designed based on the theories of circuits, Hopf bifurcation, local activity and edge of chaos. Furthermore, by utilizing the parasitic capacitor effect of the LAM, a chaotic circuit is constructed and its complex dynamics are analyzed. Finally, circuit implementations of the LAM and its oscillating circuits are demonstrated.