A mathematical analysis: From memristor to fracmemristor

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.     

Aspects of improved heat conduction relation and chemical processes in 3D Carreau fluid flow

A new fourth-order memristor chaotic oscillator is taken to investigate its fractional-order discrete synchronisation. The fractional-order differential model memristor system is transformed to its discrete model and the dynamic properties of the fractional-order discrete system are investigated. A new method for synchronising commensurate and incommensurate fractional discrete chaotic maps are proposed and validated. Numerical results are established to support the proposed methodologies. This method of synchronisation can be applied for any fractional discrete maps. Finally the fractional-order memristor system is implemented in FPGA to show that the chaotic system is hardware realisable.     

Synchronization between a novel class of fractional-order and integer-order chaotic systems via a sliding mode controller

<正>In order to figure out the dynamical behaviour of a fractional-order chaotic system and its relation to an integerorder chaotic system,in this paper we investigate the synchronization between a class of fractional-order chaotic systems and integer-order chaotic systems via sliding mode control method.Stability analysis is performed for the proposed method based on stability theorems in the fractional calculus.Moreover,three typical examples are carried out to show that the synchronization between fractional-order chaotic systems and integer-orders chaotic systems can be achieved. Our theoretical findings are supported by numerical simulation results.Finally,results from numerical computations and theoretical analysis are demonstrated to be a perfect bridge between fractional-order chaotic systems and integer-order chaotic systems.     

Firing activities in a fractional-order Hindmarsh-Rose neuron with multistable memristor as autapse

Considering the fact that memristors have the characteristics similar to biological synapses, a fractional-order multistable memristor is proposed in this paper. It is verified that the fractional-order memristor has multiple local active regions and multiple stable hysteresis loops, and the influence of fractional-order on its nonvolatility is also revealed. Then by considering the fractional-order memristor as an autapse of Hindmarsh-Rose (HR) neuron model, a fractional-order memristive neuron model is developed. The effects of the initial value, external excitation current, coupling strength and fractional-order on the firing behavior are discussed by time series, phase diagram, Lyapunov exponent and inter spike interval (ISI) bifurcation diagram. Three coexisting firing patterns, including irregular asymptotically periodic (A-periodic) bursting, A-periodic bursting and chaotic bursting, dependent on the memristor initial values, are observed. It is also revealed that the fractional-order can not only induce the transition of firing patterns, but also change the firing frequency of the neuron. Finally, a neuron circuit with variable fractional-order is designed to verify the numerical simulations.     

A general method for synchronizing an integer-order chaotic system and a fractional-order chaotic system

This paper investigates the synchronization between integer-order and fractional-order chaotic systems.By intro-ducing fractional-order operators into the controllers,the addressed problem is transformed into a synchronization one among integer-order systems.A novel general method is presented in the paper with rigorous proof.Based on this method,effective controllers are designed for the synchronization between Lorenz systems with an integer order and a fractional order,and for the synchronization between an integer-order Chen system and a fractional-order Liu system.Numerical results,which agree well with the theoretical analyses,are also given to show the effectiveness of this method.     

Comparison between two different sliding mode controllers for a fractional-order unified chaotic system

Two different sliding mode controllers for a fractional order unified chaotic system are presented. The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system, and the fractional-order system can be made asymptotically stable by this controller. By proving the existence of a sliding manifold containing fractional integral, the controller for a fractional-order system is obtained, which can stabilize it. A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.     

一个分数阶忆阻器模型及其简单串联电路的特性

忆阻器是具有时间记忆特性的非线性电阻. 经典HP TiO₂忆阻器模型的忆阻值为此前通过忆阻器电流的时间积分, 即记忆没有损失. 而最近研究证实HP TiO₂ 线性忆阻器掺杂层厚度不能等于零或者器件整体厚度, 导致器件的记忆有损失. 基于此发现, 本文首先提出了一个阶数介于0 与1间的分数阶HP TiO₂ 线性忆阻器模型, 研究了当受到周期外激励时, 分数阶导数的阶数对其忆阻值动态范围和输出电压动态幅值的影响规律, 推导出了磁滞旁瓣面积的计算公式. 结果表明, 分数阶导数阶数对磁滞回线的形状及所围成区域面积有重要影响. 特别地, 在外激频率大于1时, 分数阶忆阻器的记忆强度达到最大. 然后讨论了此分数阶忆阻器与电容或电感串联组成的单口网络的伏安特性. 结果表明, 在周期激励驱动时, 随着分数阶导数阶数的变化, 此分数阶忆阻器与电容的串联电路呈现出纯电容电路与忆阻电路的转换, 而它与电感的串联电路则呈现出纯电感电路与忆阻电路的转换.     

耦合分数阶混沌振荡器的主-从同步

近年来对分数阶系统的动力学研究得到了较为广泛的关注。本文研究了基于主-从耦合同步法的同步技术并实现了两个耦合的分数阶振荡器的混沌同步。仿真结果表明：在适当的耦合强度的调节下，该方法可实现两个耦合分数阶混沌振荡器的准确同步，且分数阶混沌振荡器的同步率明显慢于整数阶混沌振荡器的同步率；而耦合分数阶混沌振荡器在实现同步的过程中，随着阶数的提高，同步误差曲线变得平滑，这表明，系统阶数的提高改善了耦合混沌振荡器实现同步的平稳性。     

分数阶FitzHugh-Nagumo模型神经元的动力学特性及其同步

通过对分数阶FitzHugh-Nagumo模型神经元的研究,当外加电流强度作为分岔参数时,发现这种模型神经元从静息态到周期放电态所经历的Hopf分岔点不同于相应的整数阶模型神经元的分岔点;而且分数阶FitzHugh-Nagumo模型神经元呈现周期放电的外加电流强度的范围比相应的整数阶模型神经元的范围小,然而放电频率却比相应的整数阶模型神经元的放电频率高.同时还揭示在周期放电的情况下分数阶FitzHugh-Nagumo模型神经元之间的同步速率比相应的整数阶模型神经元之间的同步速率快.在数值模拟分数阶微分方程 关键词： 分数阶 Hopf分岔 FitzHugh-Nagumo模型 同步     

聚谷氨酸发酵过程中ATR-FTIR光谱信号的分数阶基线校正

采用衰减全反射傅里叶变换红外光谱法（ATR-FTIR）,结合多元校正模型对γ-聚谷氨酸（γ-PGA）发酵过程中两种主要底物葡萄糖和谷氨酸钠的浓度进行间接测量,为优化发酵系统控制提供重要的反馈信息。光谱测量中经常出现的基线漂移会严重影响后续多元校正模型的性能,需要采用基线校正算法对光谱进行预处理。现有流行的基线校正算法多数是基于Whittaker Smoother（WS）平滑算法,这些算法均采用整数阶微分对拟合基线进行约束,表达能力有限。针对现有基线校正算法中的整数阶微分自适应性差的问题,利用更加灵活的分数阶微分对基线进行约束,提出了一种基于分数阶的基线校正算法,实现对整数阶基线校正的扩展。总共进行了5个批次的γ-PGA发酵实验,并对不同批次和全部批次的ATR-FTIR光谱数据分别进行了分数阶基线校正,模型的预测精度均得到不同程度的提升。实验结果表明,只有在批次2时,基于整数阶的基线校正效果最好;其他批次的基线校正效果最好时的阶次均为分数阶。这也表明了分数阶微分（包含整数阶微分）对基线的约束更加合理。同时发现全部批次的整体基线校正效果远远差于单一批次的效果,原因可能是各批次发酵光谱的基线是不同的,对不同的批次需要选用不同的阶次以获得最佳的基线校正。此外,γ-PGA发酵样品的ATR-FTIR光谱测量是以蒸馏水为背景,会在3 100~3 600 cm-1波数范围内出现负水峰,形成有害的干扰信号;分数阶基线校正后的光谱表明,分数阶基线校正算法将负的水峰当作基线,在一定程度上进行了消除。综上分析,分数阶基线校正算法不仅扩展了传统整数阶基线校正算法的应用范围,也为消除ATR光谱中负的水峰提供了新的解决思路。     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic systems via sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. More importantly, the vector controller is one-dimensional less than the system. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Chen chaotic system and an integer-order T chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order hyperchaotic system, and the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order Lorenz chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

Continuous non-autonomous memristive Rulkov model with extreme multistability

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.     

Detecting chaos in fractional-order nonlinear systems using the smaller alignment index

The identification between chaos and ordered states in fractional-order chaotic systems is a challenge as well as a hot topic due to the complex of fractional calculus. In this paper, the smaller alignment index (SALI) is developed to detect chaos in the fractional-order chaotic systems by introducing the fractional-order tangent systems. Numerical simulations are carried out based on the fractional-order simplified Lorenz system and the fractional-order Hénon map, which are continuous chaotic system and discrete chaotic system, respectively. It shows that the proposed method is effective for distinguishing chaos and order in different kinds of fractional-order chaotic systems.     

Consensus of compound-order multi-agent systems with communication delays

In complex environments, many distributed networked systems can only be illustrated with fractional-order dynamics. When multi-agent systems show individual diversity with difference agents, heterogeneous (integer-order and fractional-order) dynamics are used to illustrate the agent systems and compose integerfractional compounded-order systems. Applying Laplace transform and frequency domain theory of the fractional-order operator, the consensus of delayed multi-agent systems with directed weighted topologies is studied. Since an integer-order model is the special case of a fractional-order model, the results in this paper can be extended to systems with integer-order models. Finally, numerical examples are used to verify our results.     

不确定分数阶时滞混沌系统自适应神经网络同步控制

针对带有完全未知的非线性不确定项和外界扰动的异结构分数阶时滞混沌系统的同步问题,基于Lyapunov稳定性理论,设计了自适应径向基函数(radial basis function,RBF)神经网络控制器以及整数阶的参数自适应律.该控制器结合了RBF神经网络和自适应控制技术,RBF神经网络用来逼近未知非线性函数,自适应律用于调整控制器中相应的参数.构造平方Lyapunov函数进行稳定性分析,基于Barbalat引理证明了同步误差渐近趋于零.数值仿真结果表明了该控制器的有效性.     

A general fractional-order dynamical network: synchronization behavior and state tuning

A general fractional-order dynamical network model for synchronization behavior is proposed. Different from previous integer-order dynamical networks, the model is made up of coupled units described by fractional differential equations, where the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. The analytic results are complemented with numerical simulations for two networks whose nodes are governed by fractional-order Lorenz dynamics and fractional-order Ro?ssler dynamics, respectively.     

Dynamic analysis and fractional-order adaptive sliding mode control for a novel fractional-order ferroresonance system

Ferroresonance is a complex nonlinear electrotechnical phenomenon, which can result in thermal and electrical stresses on the electric power system equipments due to the over voltages and over currents it generates. The prediction or determination of ferroresonance depends mainly on the accuracy of the model used. Fractional-order models are more accurate than the integer-order models. In this paper, a fractional-order ferroresonance model is proposed. The influence of the order on the dynamic behaviors of this fractional-order system under different parameters n and F is investigated.Compared with the integral-order ferroresonance system, small change of the order not only affects the dynamic behavior of the system, but also significantly affects the harmonic components of the system. Then the fractional-order ferroresonance system is implemented by nonlinear circuit emulator. Finally, a fractional-order adaptive sliding mode control(FASMC)method is used to eliminate the abnormal operation state of power system. Since the introduction of the fractional-order sliding mode surface and the adaptive factor, the robustness and disturbance rejection of the controlled system are enhanced. Numerical simulation results demonstrate that the proposed FASMC controller works well for suppression of ferroresonance over voltage.     

Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems

This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projective synchronization between three-dimensional (3D) integer-order Lorenz chaotic system and 3D fractional-order Chen chaotic system are presented to demonstrate the effectiveness of the proposed scheme.     

Exploring a hidden fermionic dark sector

In this paper, we present a generalized unified method for finding multiwave solutions of the time-fractional (2+1)-dimensional Nizhnik–Novikov–Veselov equations. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation. Multiauxiliary equations have been introduced in this method to obtain not only multisoliton solutions but also multiperiodic or multielliptic solutions. It is shown that the considered method is very effective and convenient for solving wide classes of nonlinear partial differential equations of fractional order.     

Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation

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.