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A new memristive system is presented in this paper. The peculiarity of the model is that it does not display any equilibria and exhibits periodic, chaotic, and also hyperchaotic dynamics in a particular range of the parameters space. The behavior of the proposed system is investigated through numerical simulations, such as phase portraits, Lyapunov...
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In this joint work with M.-S. Abdelouahab and L. O. Chua, we define a common mathematical framework which includes the modelling of the behavior of the new electronic circuit elements with memory recently discovered. In this general frame we extend Ohm’s law to any case.
Circuit elements that store information without the need of a power source wou...
In this paper, a class of memristive neural networks with quaternion‐valued connection weights is studied. By starting from some basic quaternion‐valued algorithms, the quaternion‐valued memristive system is obtained; then, some passivity criteria for the memristive neural networks are presented based on some appropriate auxiliary functions. Furthe...
Memristor, the missing fourth passive circuit element predicted forty years ago by Chua was recognized as a nanoscale device in 2008 by researchers of a H. P. Laboratory. Recently the notion of memristive systems was extended to capacitive and inductive elements, namely, memcapacitor and meminductor whose properties depend on the state and history...
![Figure 1. a) The SDC Carbon-based Knowm Memristor topologies [12], b)...](profile/Ertugrul-Karakulak/publication/374876710/figure/fig1/AS:11431281230100345@1710826866814/a-The-SDC-Carbon-based-Knowm-Memristor-topologies-12-b-The-SDC-Carbon-based-Knowm_Q320.jpg)
An ideal memristor that has been theoretically predicted almost a half-century ago is a nonlinear power dissipating circuit element. Nowadays, memristive systems such as thin films which are not ideal memristors are also called memristors. Such systems have current-dependent behavior and nonlinear charge-dependent electrical resistance. Self-direct...
The topics of memristive system and synchronization are two hot fields of research in nonlinear dynamics. In this paper, we introduce a memristor-based chaotic system with no equilibrium. It is found that the memristor-based system under investigation exhibits fruitful dynamic behaviors such as coexisting bifurcation, multistability, transient chao...
Citations
... Li et al. [8] proposed hyperchaotic memristive circuit by adding a quadratic ideal memristor. V.T.Pham et al. [9] proposed a Hyperchaotic system with no equilibrium point. Lai et al. [10] constructed a hyperchaotic system with no fixed points and an infinite number of coexisting attractors. ...
Chaotic or hyperchaotic systems have a significant role in engineering applications such as cryptography and secure communication, serving as primary signal generators. To ensure stronger complexity, memristors with sufficient nonlinearity are commonly incorporated into the system, suffering a limitation on the physical implementation. In this paper, we propose a new four-dimensional (4D) hyperchaotic system based on the linear memristor which is the most straightforward to implement physically. Through numerical studies, we initially demonstrate that the proposed system exhibits robust hyperchaotic behaviors under typical parameter conditions. Subsequently, we theoretically prove the existence of solid hyperchaos by combining the topological horseshoe theory with computer-assisted research. Finally, we present the realization of the proposed hyperchaotic system using an FPGA platform. This proposed system possesses two key properties. Firstly, this work suggests that the simplest memristor can also induce strong nonlinear behaviors, offering a new perspective for constructing memristive systems. Secondly, compared to existing systems, our system not only has the largest Kaplan-Yorke dimension, but also has clear advantages in areas related to engineering applications, such as the parameter range and signal bandwidth, indicating promising potential in engineering applications.
... While studying hidden attractors it was realized that existing methods are not sufficient for an abstruse understanding of them, prompting the necessity for innovative approaches within non-linear oscillation theory. Furthermore, it has been demonstrated that chaotic systems devoid of equilibrium points, featuring only stable equilibrium points or infinite equilibrium points, can exhibit hidden attractors [31][32][33][34][35][36]. Systems with hidden chaotic attractors are rarely found; then, only a few examples have been reported in the literature, and there is little knowledge about their formation, and also an analytic proof of their existence is yet needed. ...
This paper proposes a novel 4D hyperchaotic system with hidden attractors and coexisting attractors, which have no equilibrium points. The dynamic behavior of the system and five groups of coexisting attractors are analyzed by applying phase space diagrams, bifurcation diagrams and the Lyapunov exponents spectrum. Additionally, the system’s stable limit cycles and unstable periodic orbits were calculated through the variational method and then encoded using symbolic dynamics. The numerical results were verified via a circuit simulation, confirming the realizability of the novel hyperchaotic system in hardware facilities. Finally, we applied the active synchronization control method to the new system with remarkable results.
... The mathematical description of the newly designed system is as System (1) in 2014 [28], and k is the system parameter. This memristor function's complete analysis and characteristics were conducted before so that this part could be skipped. ...
... As a result of the nonlinear nature of the memristor element, the application of memristor-based circuits to build continuous chaotic or hyperchaotic systems has gained much attention in recent decades [2,11,29,34,[37][38][39]45], and analog circuits are useful in realizing continuous-time memristors [9,26]. However, digital circuits are increasingly used in practical engineering due to their easier logic programming and higher accuracy. ...
In this paper, a modified hyperchaotic memristor-based Chua's circuit and its generalized discrete model are reported. First, the dynamics of the continuous system are investigated using different tools, including stability theory, phase portraits, Lyapunov exponents, and bifurcation diagrams, showing that the proposed circuit model possesses a line of equilibrium and exhibits rich dynamics, including the coexistence of regular and chaotic attractors, and mixed-mode oscillations. Lastly, a generalized discrete version of the proposed system is formulated and its dynamics are explored. Using the Jury criterion, we constructed a useful stability test to localize some stable regions in three parameter planes. The presence of an exact periodic solution is numerically highlighted, and the occurrence of chaotic behavior is confirmed using the largest Lyapunov exponent and the 0−1 test.
... In recent years, some valuable studies have come to the fore. Pham et al. proposed a memristive hyperchaotic system without equilibrium [8]. Wang et al. introduced a memristive hyperchaotic multiscroll jerk system with controllable scroll numbers [9]. ...
PurposeThe construction of memristor-based chaotic system with complex dynamics has been a research hotspot in recent years. This paper proposes a new memristive chaotic system characterized by the abundant coexisting attractors. The new system which is established by inserting a memristor is dissipative, symmetric, chaotic and has two line equilibria.Methods
The evolution of chaos and the existence of coexisting attractors are investigated by using bifurcation diagrams and phase portraits with respect to parameters and initial conditions. Moreover, the system can realize partial amplitude control by adjusting parameters.ResultsThe new system can produce infinitely many coexisting attractors, including symmetric periodic attractors and chaotic attractors. This shows that the multistability of chaotic systems can be achieved by adding memristors.Conclusion
The analog circuit and hardware circuit are used to illustrate the existence of the proposed system. In addition, a pseudo-random number generator (PRNG) is designed based on this system and its NIST test is given. It can work reliably in the engineering environment.
... In recent years, some valuable studies have come to the fore. Pham et al. proposed a memristive hyperchaotic system without equilibrium [8]. Wang et al. introduced a memristive hyperchaotic multiscroll jerk system with controllable scroll numbers [9]. ...
Using memristors to construct special chaotic systems is highly interesting and meaningful. A simple four-dimensional memristive chaotic system with an infinite number of coexisting attractors is proposed in this paper, which has a relatively simplistic form but demonstrates complex dynamical behavior. Here, we use digital simulations to further investigate the system and using bifurcation diagrams to describe the evolution of the dynamical behavior of the system with the influence of parameters, and find that the system can generate an abundance of chaotic and periodic attractors under different parameters. The amplitude of the oscillations of the state variables of the system is closely dependent on the initial values. In addition, the experimental results of the circuit are consistent with the digital simulations, proving the existence and feasibility of this memristive chaotic system.
... Therefore, it is necessary to study a physically implementable equivalent analog circuit that exhibits memristive characteristics. In recent years, the usage of existing discrete components such as resistors, capacitors, inductors, operational amplifiers, and analog multipliers has achieved piecewise-linear memristive model [Itoh & Chua, 2008], quadratic nonlinear memristive model [Bao et al., 2010b;Guo et al., 2018], cubic nonlinear memristor-based model [Muthuswamy, 2010;Bao et al., 2010a], hyperbolic nonlinear memristive model and other types of memristor emulators [Yu et al., 2014;Corinto & Ascoli, 2012;Buscarino et al., 2012;Pham et al., 2014], which have made important contributions to the modeling analysis and experimental measurement of memristors and their application circuits. Compared with double-scroll or double-wing attractors, multi-scroll or multi-wing attractors have much more complex and abundant dynamical behaviors, much larger secret key space and better unpredictability. ...
This paper proposes a novel nonideal flux-controlled memristor model with a multipiecewise linear memductance function, which can be used to construct a memristive multi-scroll or multi-wing chaotic system. Importantly, arbitrary multi-double-scroll and multi-double-wing attractors can be generated depending on this memristor model directly and without the need to change the original nonlinear terms of the system. Another highlight is that the odd or even number of the double-scroll and double-wing attractors can also be freely controlled by the memristor model. To further illustrate these unique features, by introducing the memristor model into two classical chaotic systems, i.e. Jerk system and Lorenz system, multi-double-scroll and multi-double-wing chaotic attractors are obtained respectively. The formation mechanism of the multi-double-wing and multi-double-scroll attractors is also discussed. Moreover, the randomness of the chaotic binary sequences generated by the proposed memristor model is tested by the National Institute of Standards and Technology test suite. The tested results are better than those of the well-known Lorenz system. Furthermore, the corresponding circuits are constructed. The experimental results and the numerical simulations coincide well with each other, showing the effectiveness and feasibility of the proposed memristor model.
... ,ܣ ܤ are some functions. In this paper, we have used a memristive device as follows [40], ...
... )ݓ(ܹ is the memductance function. This function is considered as Eq. 2 [40]. )ݓ(ܹݕ݇ is the output of memristor. ...
Various chaotic systems have been studied recently. They can show many different dynamics and features. A memristive 4D chaotic oscillator with no equilibria, multistability, and hidden attractor is presented in this paper. Chaotic attractor of the proposed oscillator is discussed, and its dynamical behaviors are investigated. The oscillator does not have any equilibrium. In addition, the phenomena of multistability is studied in this system. It shows chaotic dynamics and periodic windows, verified by Lyapunov exponents’ diagram. Image encryption is studied as an engineering application of the system. The proposed system has a proper performance in encryption. Finally, this memristive chaotic system is realized using FPGA.
... Additionally, memristive devices are nonlinear and usually have hyperchaotic behaviours in electronic circuits, and hyperchaotic systems are a type of chaotic system. Similarly, memristor-based hyperchaotic circuits can have outstanding features such as multistability, extreme multistability, and hidden attractors [89]. Several parameters should be considered in designing and analysing the above-mentioned circuits based on the memristor, including approximate and not highly accurate models for the memristor such as the memristor with flux-controlled linear memductance [90], flux-controlled memductance based on a second-degree polynomial function [91,92], and piecewise linear memductance [93][94][95][96][97][98][99]. ...
This paper introduces a new hyperchaotic oscillator base on a new boundary-restricted Hewlett- Packard memristor model. Firstly, the complex system is designed based on a memristor-based hyperchaotic real system, and its properties are analyzed by means of Lyapunov exponents, Lyapunov dimension and phase portraits diagrams. Secondly, a simple feedback control based on the minimum variance control technique is designed to stabilize the hyperchaotic oscillator system, which is one of the new developed approaches for controlling the chaos in high-dimensional hyperchaotic systems. In this method, the time series variance is considered for designing and calculating the state feedback control gain. Furthermore, the state feedback control is designed so that to minimize the variance as a cost function, followed by developing an online optimization technique using the particle swarm optimization method in order to calculate the state feedback control based on the minimum variance strategy. Then, the application of this method is examined on a hyperchaotic memristor-based oscillator. Finally, the sensitivity of the proposed method is evaluated in different initial conditions that greatly influence the hyperchaotic dynamics. Considering that the optimization is online, simulation results show highly good effectiveness of the presented technique in controlling the chaos in high-dimensional hyperchaotic oscillators.
... Especially, the intrinsic nonlinear characteristic of memristor could be exploited in implementing novel chaotic systems with complex dynamics [Itoh & Chua, 2008;Muthuswamy & Chua, 2009;Muthuswamy, 2010;Iu & Fitch, 2013;Pham et al., 2014;Sabarathinam et al., 2016;Prousalis et al., 2018;Volos et al., 2018;Prousalis et al., 2019]. Memristor can replace well-known nonlinear elements, such as Chua's diode, in circuits [Itoh & Chua, 2008;Muthuswamy & Chua, 2009;Muthuswamy, 2010;Iu & Fitch, 2013] or nonlinear functions in conventional dynamical systems [Pham et al., 2014;Sabarathinam et al., 2016;Prousalis et al., 2018;Volos et al., 2018;Prousalis et al., 2019] in order to create new ones with advanced features. ...
... Especially, the intrinsic nonlinear characteristic of memristor could be exploited in implementing novel chaotic systems with complex dynamics [Itoh & Chua, 2008;Muthuswamy & Chua, 2009;Muthuswamy, 2010;Iu & Fitch, 2013;Pham et al., 2014;Sabarathinam et al., 2016;Prousalis et al., 2018;Volos et al., 2018;Prousalis et al., 2019]. Memristor can replace well-known nonlinear elements, such as Chua's diode, in circuits [Itoh & Chua, 2008;Muthuswamy & Chua, 2009;Muthuswamy, 2010;Iu & Fitch, 2013] or nonlinear functions in conventional dynamical systems [Pham et al., 2014;Sabarathinam et al., 2016;Prousalis et al., 2018;Volos et al., 2018;Prousalis et al., 2019] in order to create new ones with advanced features. Till now due to the lack of a commercially available memristor device only memristor emulators have been designed and realized conveniently for use in nonlinear chaotic circuits. ...
In the last decade, researchers, who work in the field of nonlinear circuits, have the “dream” to use a real memristor, which is the only nonlinear fundamental circuit element, in a new or other reported nonlinear circuit in literature, in order to experimentally investigate chaos. With this intention, for the first time, a well-known nonlinear circuit, in which its nonlinear element has been replaced with a commercially available memristor (KNOWM memristor), is presented in this work. Interesting phenomena concerning chaos theory, such as period-doubling route to chaos, coexisting attractors, one-scroll and double-scroll chaotic attractors are experimentally observed.
