SPECIAL TOPIC— Interdisciplinary physics: Complex network dynamics and emerging technologies

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    A review on the design of ternary logic circuits
    Xiao-Yuan Wang(王晓媛), Chuan-Tao Dong(董传涛), Zhi-Ru Wu(吴志茹), and Zhi-Qun Cheng(程知群)
    Chin. Phys. B, 2021, 30 (12): 128402.   DOI: 10.1088/1674-1056/ac248b
    Abstract490)   HTML5)    PDF (697KB)(470)      
    A multi-valued logic system is a promising alternative to traditional binary logic because it can reduce the complexity, power consumption, and area of circuit implementation. This article briefly summarizes the development of ternary logic and its advantages in digital logic circuits. The schemes, characteristics, and application of ternary logic circuits based on CMOS, CNTFET, memristor, and other devices and processes are reviewed in this paper, providing some reference for the further research and development of ternary logic circuits.
    Optimal control strategy for COVID-19 concerning both life and economy based on deep reinforcement learning
    Wei Deng(邓为), Guoyuan Qi(齐国元), and Xinchen Yu(蔚昕晨)
    Chin. Phys. B, 2021, 30 (12): 120203.   DOI: 10.1088/1674-1056/ac3229
    Abstract448)   HTML2)    PDF (998KB)(208)      
    At present, the global COVID-19 is still severe. More and more countries have experienced second or even third outbreaks. The epidemic is far from over until the vaccine is successfully developed and put on the market on a large scale. Inappropriate epidemic control strategies may bring catastrophic consequences. It is essential to maximize the epidemic restraining and to mitigate economic damage. However, the study on the optimal control strategy concerning both sides is rare, and no optimal model has been built. In this paper, the Susceptible-Infectious-Hospitalized-Recovered (SIHR) compartment model is expanded to simulate the epidemic's spread concerning isolation rate. An economic model affected by epidemic isolation measures is established. The effective reproduction number and the eigenvalues at the equilibrium point are introduced as the indicators of controllability and stability of the model and verified the effectiveness of the SIHR model. Based on the Deep Q Network (DQN), one of the deep reinforcement learning (RL) methods, the blocking policy is studied to maximize the economic output under the premise of controlling the number of infections in different stages. The epidemic control strategies given by deep RL under different learning strategies are compared for different reward coefficients. The study demonstrates that optimal policies may differ in various countries depending on disease spread and anti-economic risk ability. The results show that the more economical strategy, the less economic loss in the short term, which can save economically fragile countries from economic crises. In the second or third outbreak stage, the earlier the government adopts the control strategy, the smaller the economic loss. We recommend the method of deep RL to specify a policy which can control the epidemic while making quarantine economically viable.
    Prediction of epidemics dynamics on networks with partial differential equations: A case study for COVID-19 in China
    Ru-Qi Li(李汝琦), Yu-Rong Song(宋玉蓉), and Guo-Ping Jiang(蒋国平)
    Chin. Phys. B, 2021, 30 (12): 120202.   DOI: 10.1088/1674-1056/ac2b16
    Abstract478)   HTML2)    PDF (799KB)(133)      
    Since December 2019, the COVID-19 epidemic has repeatedly hit countries around the world due to various factors such as trade, national policies and the natural environment. To closely monitor the emergence of new COVID-19 clusters and ensure high prediction accuracy, we develop a new prediction framework for studying the spread of epidemic on networks based on partial differential equations (PDEs), which captures epidemic diffusion along the edges of a network driven by population flow data. In this paper, we focus on the effect of the population movement on the spread of COVID-19 in several cities from different geographic regions in China for describing the transmission characteristics of COVID-19. Experiment results show that the PDE model obtains relatively good prediction results compared with several typical mathematical models. Furthermore, we study the effectiveness of intervention measures, such as traffic lockdowns and social distancing, which provides a new approach for quantifying the effectiveness of the government policies toward controlling COVID-19 via the adaptive parameters of the model. To our knowledge, this work is the first attempt to apply the PDE model on networks with Baidu Migration Data for COVID-19 prediction.
    Enhance sensitivity to illumination and synchronization in light-dependent neurons
    Ying Xie(谢盈), Zhao Yao(姚昭), Xikui Hu(胡锡奎), and Jun Ma(马军)
    Chin. Phys. B, 2021, 30 (12): 120510.   DOI: 10.1088/1674-1056/ac1fdc
    Abstract477)   HTML5)    PDF (3491KB)(135)      
    When a phototube is activated to connect a neural circuit, the output voltage becomes sensitive to external illumination because the photocurrent across the phototube can be controlled by external electromagnetic wave. The channel currents from different branch circuits have different impacts on the outputs voltage of the neural circuit. In this paper, a phototube is incorporated into different branch circuits in a simple neural circuit, and then a light-controlled neuron is obtained for further nonlinear analysis. Indeed, the phototube is considered as exciting source when it is activated by external illumination, and two kinds of light-sensitive neurons are obtained when the phototube is connected to capacitor or induction coil, respectively. Electric synapse coupling is applied to detect possible synchronization between two functional neurons, and the energy consumption along the coupling channel via resistor is estimated. The analog circuits for the two kinds of light-sensitive neurons are supplied for further confirmation by using Multisim. It is found that two light-sensitive neurons and neural circuits can be synchronized by taming the coupling intensity carefully. It provides possible clues to understand the synchronization mechanism for eyes and artificial sensors which are sensitive to illumination. Finally, a section for open problems is supplied for further investigation about its collective behaviors in the network with/without synapse coupling.
    Embedding any desired number of coexisting attractors in memristive system
    Chunbiao Li(李春彪), Ran Wang(王然), Xu Ma(马旭), Yicheng Jiang(姜易成), and Zuohua Liu(刘作华)
    Chin. Phys. B, 2021, 30 (12): 120511.   DOI: 10.1088/1674-1056/ac1e1f
    Abstract377)   HTML2)    PDF (4897KB)(259)      
    A simple variable-boostable system is selected as the structure for hosting an arbitrarily defined memristor for chaos producing. The derived three-dimensional (3-D) memristive chaotic system shows its distinct property of offset, amplitude and frequency control. Owing its merits any desired number of coexisting attractors are embedded by means of attractor doubling and self-reproducing based on function-oriented offset boosting. In this circumstance two classes of control gates are found:one determines the number of coexisting attractors resorting to the independent offset controller while the other is the initial condition selecting any one of them. Circuit simulation gives a consistent output with theoretically predicted embedded attractors.
    Controlling chaos and supressing chimeras in a fractional-order discrete phase-locked loop using impulse control
    Karthikeyan Rajagopal, Anitha Karthikeyan, and Balamurali Ramakrishnan
    Chin. Phys. B, 2021, 30 (12): 120512.   DOI: 10.1088/1674-1056/ac1b83
    Abstract389)   HTML0)    PDF (6222KB)(141)      
    A fractional-order difference equation model of a third-order discrete phase-locked loop (FODPLL) is discussed and the dynamical behavior of the model is demonstrated using bifurcation plots and a basin of attraction. We show a narrow region of loop gain where the FODPLL exhibits quasi-periodic oscillations, which were not identified in the integer-order model. We propose a simple impulse control algorithm to suppress chaos and discuss the effect of the control step. A network of FODPLL oscillators is constructed and investigated for synchronization behavior. We show the existence of chimera states while transiting from an asynchronous to a synchronous state. The same impulse control method is applied to a lattice array of FODPLL, and the chimera states are then synchronized using the impulse control algorithm. We show that the lower control steps can achieve better control over the higher control steps.
    Sensitivity to external optical feedback of circular-side hexagonal resonator microcavity laser
    Tong Zhao(赵彤), Zhi-Ru Shen(申志儒), Wen-Li Xie(谢文丽), Yan-Qiang Guo(郭龑强), An-Bang Wang(王安帮), and Yun-Cai Wang(王云才)
    Chin. Phys. B, 2021, 30 (12): 120513.   DOI: 10.1088/1674-1056/ac322a
    Abstract383)   HTML0)    PDF (1766KB)(98)      
    The sensitivity to fault reflection is very important for larger dynamic range in fiber fault detection technique. Using time delay signature (TDS) of chaotic laser formed by optical feedback can solve the sensitivity limitation of photodetector in fiber fault detection. The TDS corresponds to the feedback position and the fault reflection can be detected by the laser diode. The sensitivity to feedback level of circular-side hexagonal resonator (CSHR) microcavity laser is numerically simulated and the feedback level boundaries of each output dynamic state are demonstrated. The peak level of TDS is utilized to analyze the sensitivity. The demonstration is presented in two aspects:the minimum feedback level when the TDS emerges and the variation degree of TDS level on feedback level changing. The results show that the CSHR microcavity laser can respond to the feedback level of 0.07%, corresponding to -63-dB feedback strength. Compared to conventional distributed feedback laser, the sensitivity improves almost 20 dB due to the shorter internal cavity length of CSHR microcavity laser. Moreover, 1% feedback level changing will induce 1.001 variation on TDS level, and this variation degree can be influenced by other critical internal parameters (active region side length, damping rate, and linewidth enhancement factor).
    Cascade discrete memristive maps for enhancing chaos
    Fang Yuan(袁方), Cheng-Jun Bai(柏承君), and Yu-Xia Li(李玉霞)
    Chin. Phys. B, 2021, 30 (12): 120514.   DOI: 10.1088/1674-1056/ac20c7
    Abstract511)   HTML0)    PDF (3433KB)(135)      
    Continuous-time memristor (CM) has been widely used to generate chaotic oscillations. However, discrete memristor (DM) has not been received adequate attention. Motivated by the cascade structure in electronic circuits, this paper introduces a method to cascade discrete memristive maps for generating chaos and hyperchaos. For a discrete-memristor seed map, it can be self-cascaded many times to get more parameters and complex structures, but with larger chaotic areas and Lyapunov exponents. Comparisons of dynamic characteristics between the seed map and cascading maps are explored. Meanwhile, numerical simulation results are verified by the hardware implementation.
    Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation
    Zong-Li Yang(杨宗立), Dong Liang(梁栋), Da-Wei Ding(丁大为), Yong-Bing Hu(胡永兵), and Hao Li(李浩)
    Chin. Phys. B, 2021, 30 (12): 120515.   DOI: 10.1088/1674-1056/ac1fdf
    Abstract412)   HTML1)    PDF (9816KB)(138)      
    This paper proposes a fractional-order simplest chaotic system using a bi-stable locally-active memristor. The characteristics of the memristor and transient transition behaviors of the proposed system are analyzed, and this circuit is implemented digitally using ARM-based MCU. Firstly, the mathematical model of the memristor is designed, which is nonvolatile, locally-active and bi-stable. Secondly, the asymptotical stability of the fractional-order memristive chaotic system is investigated and some sufficient conditions of the stability are obtained. Thirdly, complex dynamics of the novel system are analyzed using phase diagram, Lyapunov exponential spectrum, bifurcation diagram, basin of attractor, and coexisting bifurcation, coexisting attractors are observed. All of these results indicate that this simple system contains the abundant dynamic characteristics. Moreover, transient transition behaviors of the system are analyzed, and it is found that the behaviors of transient chaotic and transient period transition alternately occur. Finally, the hardware implementation of the fractional-order bi-stable locally-active memristive chaotic system using ARM-based STM32F750 is carried out to verify the numerical simulation results.
    Modeling and dynamics of double Hindmarsh-Rose neuron with memristor-based magnetic coupling and time delay
    Guoyuan Qi(齐国元) and Zimou Wang(王子谋)
    Chin. Phys. B, 2021, 30 (12): 120516.   DOI: 10.1088/1674-1056/ac16cc
    Abstract502)   HTML0)    PDF (4128KB)(187)      
    The firing of a neuron model is mainly affected by the following factors:the magnetic field, external forcing current, time delay, etc. In this paper, a new time-delayed electromagnetic field coupled dual Hindmarsh-Rose neuron network model is constructed. A magnetically controlled threshold memristor is improved to represent the self-connected and the coupled magnetic fields triggered by the dynamic change of neuronal membrane potential for the adjacent neurons. Numerical simulation confirms that the coupled magnetic field can activate resting neurons to generate rich firing patterns, such as spiking firings, bursting firings, and chaotic firings, and enable neurons to generate larger firing amplitudes. The study also found that the strength of magnetic coupling in the neural network also affects the number of peaks in the discharge of bursting firing. Based on the existing medical treatment background of mental illness, the effects of time lag in the coupling process against neuron firing are studied. The results confirm that the neurons can respond well to external stimuli and coupled magnetic field with appropriate time delay, and keep periodic firing under a wide range of external forcing current.
    Modeling the dynamics of firms' technological impact
    Shuqi Xu(徐舒琪), Manuel Sebastian Mariani, and Linyuan Lü(吕琳媛)
    Chin. Phys. B, 2021, 30 (12): 120517.   DOI: 10.1088/1674-1056/ac364c
    Abstract529)   HTML9)    PDF (1832KB)(176)      
    Recent studies in complexity science have uncovered temporal regularities in the dynamics of impact along scientific and other creative careers, but they did not extend the obtained insights to firms. In this paper, we show that firms' technological impact patterns cannot be captured by the state-of-the-art dynamical models for the evolution of scientists' research impact, such as the Q model. Therefore, we propose a time-varying returns model which integrates the empirically-observed relation between patent order and technological impact into the Q model. The proposed model can reproduce the timing pattern of firms' highest-impact patents accurately. Our results shed light on modeling the differences behind the impact dynamics of researchers and firms.
    Continuous non-autonomous memristive Rulkov model with extreme multistability
    Quan Xu(徐权), Tong Liu(刘通), Cheng-Tao Feng(冯成涛), Han Bao(包涵), Hua-Gan Wu(武花干), and Bo-Cheng Bao(包伯成)
    Chin. Phys. B, 2021, 30 (12): 128702.   DOI: 10.1088/1674-1056/ac2f30
    Abstract482)   HTML3)    PDF (1676KB)(199)      
    Based on the two-dimensional (2D) discrete Rulkov model that is used to describe neuron dynamics, this paper presents a continuous non-autonomous memristive Rulkov model. The effects of electromagnetic induction and external stimulus are simultaneously considered herein. The electromagnetic induction flow is imitated by the generated current from a flux-controlled memristor and the external stimulus is injected using a sinusoidal current. Thus, the presented model possesses a line equilibrium set evolving over the time. The equilibrium set and their stability distributions are numerically simulated and qualitatively analyzed. Afterwards, numerical simulations are executed to explore the dynamical behaviors associated to the electromagnetic induction, external stimulus, and initial conditions. Interestingly, the initial conditions dependent extreme multistability is elaborately disclosed in the continuous non-autonomous memristive Rulkov model. Furthermore, an analog circuit of the proposed model is implemented, upon which the hardware experiment is executed to verify the numerically simulated extreme multistability. The extreme multistability is numerically revealed and experimentally confirmed in this paper, which can widen the future engineering employment of the Rulkov model.
    Explosive synchronization: From synthetic to real-world networks
    Atiyeh Bayani, Sajad Jafari, and Hamed Azarnoush
    Chin. Phys. B, 2022, 31 (2): 020504.   DOI: 10.1088/1674-1056/ac3cb0
    Abstract410)   HTML3)    PDF (12796KB)(256)      
    Synchronization is a widespread phenomenon in both synthetic and real-world networks. This collective behavior of simple and complex systems has been attracting much research during the last decades. Two different routes to synchrony are defined in networks; first-order, characterized as explosive, and second-order, characterized as continuous transition. Although pioneer researches explained that the transition type is a generic feature in the networks, recent studies proposed some frameworks in which different phase and even chaotic oscillators exhibit explosive synchronization. The relationship between the structural properties of the network and the dynamical features of the oscillators is mainly proclaimed because some of these frameworks show abrupt transitions. Despite different theoretical analyses about the appearance of the first-order transition, studies are limited to the mean-field theory, which cannot be generalized to all networks. There are different real-world and man-made networks whose properties can be characterized in terms of explosive synchronization, e.g., the transition from unconsciousness to wakefulness in the brain and spontaneous synchronization of power-grid networks. In this review article, explosive synchronization is discussed from two main aspects. First, pioneer articles are categorized from the dynamical-structural framework point of view. Then, articles that considered different oscillators in the explosive synchronization frameworks are studied. In this article, the main focus is on the explosive synchronization in networks with chaotic and neuronal oscillators. Also, efforts have been made to consider the recent articles which proposed new frameworks of explosive synchronization.
    FPGA implementation and image encryption application of a new PRNG based on a memristive Hopfield neural network with a special activation gradient
    Fei Yu(余飞), Zinan Zhang(张梓楠), Hui Shen(沈辉), Yuanyuan Huang(黄园媛), Shuo Cai(蔡烁), and Sichun Du(杜四春)
    Chin. Phys. B, 2022, 31 (2): 020505.   DOI: 10.1088/1674-1056/ac3cb2
    Abstract441)   HTML2)    PDF (11131KB)(306)      
    A memristive Hopfield neural network (MHNN) with a special activation gradient is proposed by adding a suitable memristor to the Hopfield neural network (HNN) with a special activation gradient. The MHNN is simulated and dynamically analyzed, and implemented on FPGA. Then, a new pseudo-random number generator (PRNG) based on MHNN is proposed. The post-processing unit of the PRNG is composed of nonlinear post-processor and XOR calculator, which effectively ensures the randomness of PRNG. The experiments in this paper comply with the IEEE 754-1985 high precision 32-bit floating point standard and are done on the Vivado design tool using a Xilinx XC7Z020CLG400-2 FPGA chip and the Verilog-HDL hardware programming language. The random sequence generated by the PRNG proposed in this paper has passed the NIST SP800-22 test suite and security analysis, proving its randomness and high performance. Finally, an image encryption system based on PRNG is proposed and implemented on FPGA, which proves the value of the image encryption system in the field of data encryption connected to the Internet of Things (IoT).
    Energy spreading, equipartition, and chaos in lattices with non-central forces
    Arnold Ngapasare, Georgios Theocharis, Olivier Richoux, Vassos Achilleos, and Charalampos Skokos
    Chin. Phys. B, 2022, 31 (2): 020506.   DOI: 10.1088/1674-1056/ac3a5e
    Abstract375)   HTML4)    PDF (3847KB)(89)      
    We numerically study a one-dimensional, nonlinear lattice model which in the linear limit is relevant to the study of bending (flexural) waves. In contrast with the classic one-dimensional mass-spring system, the linear dispersion relation of the considered model has different characteristics in the low frequency limit. By introducing disorder in the masses of the lattice particles, we investigate how different nonlinearities in the potential (cubic, quadratic, and their combination) lead to energy delocalization, equipartition, and chaotic dynamics. We excite the lattice using single site initial momentum excitations corresponding to a strongly localized linear mode and increase the initial energy of excitation. Beyond a certain energy threshold, when the cubic nonlinearity is present, the system is found to reach energy equipartition and total delocalization. On the other hand, when only the quartic nonlinearity is activated, the system remains localized and away from equipartition at least for the energies and evolution times considered here. However, for large enough energies for all types of nonlinearities we observe chaos. This chaotic behavior is combined with energy delocalization when cubic nonlinearities are present, while the appearance of only quadratic nonlinearity leads to energy localization. Our results reveal a rich dynamical behavior and show differences with the relevant Fermi-Pasta-Ulam-Tsingou model. Our findings pave the way for the study of models relevant to bending (flexural) waves in the presence of nonlinearity and disorder, anticipating different energy transport behaviors.
    Corrigendum to “The transition from conservative to dissipative flows in class-B laser model with fold-Hopf bifurcation andcoexisting attractors”
    Yue Li(李月), Zengqiang Chen(陈增强), Mingfeng Yuan(袁明峰), and Shijian Cang(仓诗建)
    Chin. Phys. B, 2023, 32 (6): 069902.   DOI: 10.1088/1674-1056/acd8a6
    Abstract346)   HTML5)    PDF (618KB)(169)      
    Recently, we received a letter from Prof. G. L. Oppo, which indicated that he had doubts about the transformation of the system in the article Chin. Phys. B 31 060503 (2022) and gave other considerations. After inspection, we found that there was a clerical error in the article. Based on this, we have made corrections and supplements to the original article.
    A novel similarity measure for mining missing links in long-path networks
    Yijun Ran(冉义军), Tianyu Liu(刘天宇), Tao Jia(贾韬), and Xiao-Ke Xu(许小可)
    Chin. Phys. B, 2022, 31 (6): 068902.   DOI: 10.1088/1674-1056/ac4483
    Abstract539)   HTML1)    PDF (5020KB)(130)      
    Network information mining is the study of the network topology, which may answer a large number of application-based questions towards the structural evolution and the function of a real system. The question can be related to how the real system evolves or how individuals interact with each other in social networks. Although the evolution of the real system may seem to be found regularly, capturing patterns on the whole process of evolution is not trivial. Link prediction is one of the most important technologies in network information mining, which can help us understand the evolution mechanism of real-life network. Link prediction aims to uncover missing links or quantify the likelihood of the emergence of nonexistent links from known network structures. Currently, widely existing methods of link prediction almost focus on short-path networks that usually have a myriad of close triangular structures. However, these algorithms on highly sparse or long-path networks have poor performance. Here, we propose a new index that is associated with the principles of structural equivalence and shortest path length (SESPL) to estimate the likelihood of link existence in long-path networks. Through a test of 548 real networks, we find that SESPL is more effective and efficient than other similarity-based predictors in long-path networks. Meanwhile, we also exploit the performance of SESPL predictor and of embedding-based approaches via machine learning techniques. The results show that the performance of SESPL can achieve a gain of 44.09% over GraphWave and 7.93% over Node2vec. Finally, according to the matrix of maximal information coefficient (MIC) between all the similarity-based predictors, SESPL is a new independent feature in the space of traditional similarity features.
    Advantage of populous countries in the trends of innovation efficiency
    Dan-Dan Hu(胡淡淡), Xue-Jin Fang(方学进), and Xiao-Pu Han(韩筱璞)
    Chin. Phys. B, 2022, 31 (6): 068903.   DOI: 10.1088/1674-1056/ac5614
    Abstract562)   HTML1)    PDF (3254KB)(88)      
    A flurry of studies indicates that population size has a positive effect on innovation, however, cross-country empirical evidence remains sparse. In this paper, we add to the literature by investigating the relationship between population size and innovation efficiency at the country level through constructing three relative indexes based on the datasets of patent applications and Research and Development (R&D) investment. Different from previous studies based on absolute innovation indicators, the relative indexes can reflect the core innovation efficiency of economies by excluding the impact from the difference of economic development level, with a view putting all economies into a comparable standard framework. For all of the three relative indexes, their long-term trends show significant correlations with population size, and the economy with a larger population usually has better and stable performance on the trends of innovation efficiency. In addition, we find that there is a critical population size, over which the economy would be more likely to have a spontaneous improvement on innovation efficiency. This study provides direct evidence in supporting the population size advantage on the trends of innovation efficiency at the economy level and provides new insight to understand the rapid development of innovation in a few populous countries.
    The dynamics of a memristor-based Rulkov neuron with fractional-order difference
    Yan-Mei Lu(卢艳梅), Chun-Hua Wang(王春华), Quan-Li Deng(邓全利), and Cong Xu(徐聪)
    Chin. Phys. B, 2022, 31 (6): 060502.   DOI: 10.1088/1674-1056/ac539a
    Abstract352)   HTML2)    PDF (6239KB)(134)      
    The exploration of the memristor model in the discrete domain is a fascinating hotspot. The electromagnetic induction on neurons has also begun to be simulated by some discrete memristors. However, most of the current investigations are based on the integer-order discrete memristor, and there are relatively few studies on the form of fractional order. In this paper, a new fractional-order discrete memristor model with prominent nonlinearity is constructed based on the Caputo fractional-order difference operator. Furthermore, the dynamical behaviors of the Rulkov neuron under electromagnetic radiation are simulated by introducing the proposed discrete memristor. The integer-order and fractional-order peculiarities of the system are analyzed through the bifurcation graph, the Lyapunov exponential spectrum, and the iterative graph. The results demonstrate that the fractional-order system has more abundant dynamics than the integer one, such as hyper-chaos, multi-stable and transient chaos. In addition, the complexity of the system in the fractional form is evaluated by the means of the spectral entropy complexity algorithm and consequences show that it is affected by the order of the fractional system. The feature of fractional difference lays the foundation for further research and application of the discrete memristor and the neuron map in the future.
    A mathematical analysis: From memristor to fracmemristor
    Wu-Yang Zhu(朱伍洋), Yi-Fei Pu(蒲亦非), Bo Liu(刘博), Bo Yu(余波), and Ji-Liu Zhou(周激流)
    Chin. Phys. B, 2022, 31 (6): 060204.   DOI: 10.1088/1674-1056/ac615c
    Abstract541)   HTML4)    PDF (1521KB)(222)      
    The memristor is also a basic electronic component, just like resistors, capacitors and inductors. It is a nonlinear device with memory characteristics. In 2008, with HP's announcement of the discovery of the TiO2 memristor, the new memristor system, memory capacitor (memcapacitor) and memory inductor (meminductor) were derived. Fractional-order calculus has the characteristics of non-locality, weak singularity and long term memory which traditional integer-order calculus does not have, and can accurately portray or model real-world problems better than the classic integer-order calculus. In recent years, researchers have extended the modeling method of memristor by fractional calculus, and proposed the fractional-order memristor, but its concept is not unified. This paper reviews the existing memristive elements, including integer-order memristor systems and fractional-order memristor systems. We analyze their similarities and differences, give the derivation process, circuit schematic diagrams, and an outlook on the development direction of fractional-order memristive elements.