Special Issue Editorial: “Discrete and Continuous Memristive Nonlinear Systems and Symmetry”

Memristor, as the fourth basic electronic component, was first reported by Chua in 1971 [1]. As we all know, the other three basic electronic components are resistance, capacitance and inductance, and they are widely used in the engineering field. In 2008, the HP (Hewlett-Packard) Laboratory realized this component using materialogy [2]. It indicated the physical realizability of memristor. Since then, the design and simulation of memristors has become a hot topic. However, due to the lack of commercial memristor products, they have not been widely used in practical applications. According to the current research frontiers, it is not difficult to find that memristors have potential application value in fields such as brain-like computing, storage and nonlinear systems. For instance, by introducing memristors to the nonlinear systems, rich dynamics including multistability, hidden attractors can be found. The discrete mathematical modelling has been discussed previously, and it has been proved that the discrete memristor can be used to design new chaotic maps [3]. Meanwhile, a fractional-order memristor has also been designed; it shows a different memory effect due to the different kernel function in the fractional calculus [4]. Since there is nonlinearity, the design of both discrete and continuous memristive nonlinear systems with symmetry is a new research hotspot.

As a result, in this Special Issue, both continues and discrete memristor-based nonlinear systems and their applications are reported. For continuous nonlinear systems, the Chen memristor chaotic system [5], the Jerk memristor chaotic system [6] and the memristor circuit based on its van der Pol oscillator [7] are designed. For memristive chaotic maps, a new product trigonometric chaotic map [8], a discrete memristor dimension-changeable hyperchaotic map [9] and a general model for constructing memristive maps [10] are discussed. Meanwhile, memristor-based applications are considered. An audio encryption algorithm [5], a multi-channel data aggregation scheduling algorithm [11], a K-means clustering algorithm [12] and a ferroelectric memristor-based transient chaotic neural network [13] are designed. Moreover, the mild solutions for fractional impulsive integro-differential evolution equations are investigated [14]. Although this is not about memristive systems, the issue of fractional numerical solutions is discussed. Finally, the published articles were prepared by scientists working at leading universities and research centres in China, India, Vietnam, Jordan and Saudi Arabia. However, in the second volume, more researchers from outside of China should be attracted.

W. Dai et al. [5], in the paper “Audio Encryption Algorithm Based on Chen Memristor Chaotic System”, proposed an audio encryption algorithm based on a Chen memristor chaotic system, where the fast Walsh–Hadamar Transform (FWHT) is employed to compress and denoise the signal. They show that, by introducing the memristor to the Chen system, the periodic window problems, including the limited chaos range and nonuniform distribution, are solved. It also indicates that, compared with other audio encryption methods, the one proposed in this method is greatly improved.

X. Wu et al. [6], in the paper “From Memristor-Modeled Jerk System to the Nonlinear Systems with Memristor”, discussed the difference between nonlinear systems with memristor function and memristor electronic components. Firstly, a memristor-modeled Jerk system is proposed, and dynamics analysis and circuit implementation are carried out. It shows that the so-called memristor chaotic systems are not all nonlinear system with memristor electronic components. There are some rules to distinguish these two types of systems. Finally, nonlinear systems with memristors are reviewed.

B. Yang et al. [7], in the paper “Symplectic Dynamics and Simultaneous Resonance Analysis of Memristor Circuit Based on Its van der Pol Oscillator”, designed a non-autonomous memristor circuit based on van der Pol oscillator. Different numerical simulation algorithms, including the Euler method, symplectic Euler method, four-order Runge–Kutta method, four-order symplectic Runge–Kutta–Nyström method and multi-scale method, are compared when used to solve the systems. Finally, the chaotic dynamical behaviors versus system parameters are analyzed. According to this manuscript, this provides a deeper understanding of the oscillation characteristics of other nonlinear oscillation problems in the form of the van der Pol equation in the future.

Q. Lu et al. [8], in the paper “Symmetric Image Encryption Algorithm Based on a New Product Trigonometric Chaotic Map”, designed a product trigonometric map by introducing a cosine function into the Sine chaotic map. The complex dynamics were analyzed using the ApEn algorithm, Cobweb Graph and NIST test. The randomness of the generated time series was shown. A symmetric image encryption algorithm is designed where the key is related to the hash value of the image. According to the security analysis and comparisons with other methods reported in the literature, it shows a good cryptographic performance and high time efficiency of the proposed method.

C. Wei et al. [9], in the paper “Design of a New Dimension-Changeable Hyperchaotic Model Based on Discrete Memristor”, proposed a higher-dimensional discrete memristior hyperchaotic scheme using a closed-loop coupling cascade operation. The proposed scheme is dimension-changeable. The Sine map is used as the scheme for modulation purpose and the Logistic map is used as the internal disturbance. The dynamics of the proposed system are analyzed by means of Lyapunov exponents, bifurcation diagram, complexity and the 0–1 test. It is shown that the discrete memristor can effectively improve the performance of the discrete chaotic map and make the hyperchaotic system more stable.

J. Ramadoss et al. [10], in the paper “Discrete Memristance and Nonlinear Term for Designing Memristive Maps”, proposed a general structure to design discrete memristive chaotic maps and a class of chaotic maps connected in serial with a nonlinear term and a discrete memristor are obtained. The dynamics of the example systems are analyzed by means of fixed points, bifurcation diagram, symmetry, and coexisting iterative plots.

Y. Lu et al. [11], in the paper “Multi-Channel Data Aggregation Scheduling Based on the Chaotic Firework Algorithm for the Battery-Free Wireless Sensor Network”, proposed a multi-channel data aggregation scheduling method based on the chaotic firework algorithm for a battery-free wireless sensor network, where the discrete memristor-based Hénon map is used to generated chaotic time series. The new method can coordinate the communication time and occupy channels with a large number of sensor nodes. According to the simulation result, it shows the advantages of aggregation delay and occupied channels in the new scheduling method.

Y. Wan et al. [12], in the paper “K-Means Clustering Algorithm Based on Memristive Chaotic System and Sparrow Search Algorithm”, proposed a K-means clustering algorithm for data clustering based on the memristive chaotic sparrow search algorithm. In this work, the memristive chaotic sparrow search algorithm is applied to find the optimal locations as the initial cluster centroids for the K-means algorithm. The proposed method is employed to analyze college students’ academic data, and is found to be effective.

Z. Lin et al. [13], in the paper “A Ferroelectric Memristor-Based Transient Chaotic Neural Network for Solving Combinatorial Optimization Problems”, proposed a method to use a ferroelectric memristor to implement the annealing function of a transient chaotic neural network. In this proposed network, the self-feedback connection weight is designed using the conductance of the ferroelectric memristor and it can be dynamically adjusted. As a result, a ferroelectric memristor-based transient chaotic neural network is designed and it is employed to solve the traveling salesman problem. According to the 1000 runs for the 10-city traveling salesman problem, the designed network achieves a shorter average path distance, highlighting its effectiveness.

Y. Li et al. [14], in the paper “Mild Solutions for Fractional Impulsive Integro-Differential Evolution Equations with Nonlocal Conditions in Banach Spaces”, proposed an approach to obtain mild solutions for fractional impulsive integro-differential evolution equations by using the cosine family theory, measure of non-compactness and the Mönch fixed-point theorem. In the target system, the derivative order is $1<\beta \le 2$, and the given example is presented to demonstrate the obtained results.

In conclusion, in this Special Issue, we focus on interdisciplinary science by considering the discrete and continuous memristive nonlinear systems with or without fractional calculus. They can be applied in different research fields including nonlinear physics, artificial intelligence, mathematics and engineering. We hope that, through this Special Issue, more scholars will be motivated to forward research on the applications of memristors, especially discrete memristors.