Hidden Extreme Multistability Research Articles

Hidden extreme multistability in a smooth flux-controlled memristor based four-dimensional chaotic system and its application in image encryption

Abstract This paper introduces and investigates the dynamic analysis of a four-dimensional chaotic system based on a flux-controlled memristor and its application in image encryption. The analysis of equilibrium points of the system shows that it has no equilibrium points and therefore belongs to the family of systems with hidden attractors. It is demonstrated by computational simulations that the system under study experiences rich and complicated dynamics such as coexisting hidden attractors, hidden extreme multistability, offset boosting dynamics and antimonotonicity. A physical circuit that can reproduce the complex behaviors of the system is constructed and realized. The results are in agreement with those from computational simulations. According to the interesting features of the system, it is exploited to generate random bits for image encryption. The robustness and efficiency of the designed image encryption algorithm are tested by providing some statistical tests and security performances. It is found that the designed image encryption algorithm is efficient and robust compared to other image encryption algorithms investigated in some other recent related works.

A novel 5D memristor conservative chaotic system with multiple forms of hidden flows

Memristor is one of the basic circuit elements commonly used in circuit model analysis. More complex dynamic characteristics can be observed by coupling memristor into nonlinear circuit. However, there is relatively little attention paid to high-dimensional conservative chaos based on memristors up to now. In this paper, a five-dimensional memristor conservative chaotic system is built after the introduction of the memristor into conservative chaotic system. There is no equilibrium point in this system and the phase trajectory produced by it has hidden properties. Its conservatism is analyzed by bifurcation diagram, Lyapunov exponent spectrum and divergence. The phase trajectory will change with the change of parameters, which Poincaré mapping also verified these dynamic behaviors. In addition, hidden extreme multistability and initial value offset boosting behavior are also found in this system. It is to be noted that this behavior is less in memristor conservative chaotic system without equilibrium points. At the same time, a new transient transition behavior is observed. By introducing spectral entropy algorithm, the complexity of sequences is analyzed and compared with the existing literature. The results show that the system has higher complexity. Finally, the systematic analogous circuit is designed and built whose results are consistent with the MATLAB numerical simulation results, which has laid a solid foundation for the practical application of the system in engineering.

A 4D chaotic system with four-wing attractors and hidden extreme multistability

In this paper, a novel four-wing chaotic system is constructed based on the Sprott-A system. The novel system contains three nonlinearly quadratic terms, which have rich dynamics such as four-wing hidden attractors, hidden extreme multistability and transient transitions. It is found that this system has high complexity by spectral entropy analysis. In addition, a corresponding hardware analog circuit is designed based on this system by operational amplifiers and multipliers. The experimental results agree with those of the theoretical analysis and confirm that this system is practically feasible. This system will have a good application value in secure communication and cryptography.

Hidden extreme multistability and synchronicity of memristor-coupled non-autonomous memristive Fitzhugh–Nagumo models

Hidden extreme multistability and synchronicity of memristor-coupled non-autonomous memristive Fitzhugh–Nagumo models

Memristor hyperchaos in a generalized Kolmogorov-type system with extreme multistability

Memristor chaotic systems have aroused great attention in recent years with their potentials expected in engineering applications. In this paper, a five-dimension (5D) double-memristor hyperchaotic system (DMHS) is modeled by introducing two active magnetron memristor models into the Kolmogorov-type formula. The boundness condition of the proposed hyperchaotic system is proved. Coexisting bifurcation diagram and numerical verification explain the bistability. The rich dynamics of the system are demonstrated by the dynamic evolution map and the basin. The simulation results reveal the existence of transient hyperchaos and hidden extreme multistability in the presented DMHS. The NIST tests show that the generated signal sequence is highly random, which is feasible for encryption purposes. Furthermore, the system is implemented based on a FPGA experimental platform, which benefits the further applications of the proposed hyperchaos.

Hidden extreme multistability and its control with selection of a desired attractor in a non-autonomous Hopfield neuron

In this article, a single nonautonomous Hopfield neuron is investigated. This neuron involves a new hyperbolic-type memristor as self-synaptic weight. The current–voltage characteristics of the new memristor enables the observation of the hysteresis loop with more than one pinch. The analysis of the equilibria highlighted that the model has hidden dynamics. A plethora of nonlinear phenomena is obtained during the analysis of the model. Among them, the coexistence of an infinite number of symmetric bifurcation diagrams each having its intrinsic energetic singularity. The zooms of the initial condition diagrams enable to observe the coexistence of ten periodic hidden attractors and nine chaotic hidden attractors. The model also exhibited the antimonotonicity phenomenon characterized by the coexistence of an infinite number of bubbles. The control of up to nine coexisting hidden chaotic attractors in a specific domain of the initial condition is successfully realized. This control is achieved by choosing initially an attractor as a designated survivor using the feedback term method. It is then found that the employed method can also be used to control multistability in a system without an equilibrium point. An analog circuit of the model is also designed in PSpice, to further support the results of the theoretical investigations.

Memristor-based time-delay chaotic system with hidden extreme multi-stability and pseudo-random sequence generator

Memristor-based time-delay chaotic system with hidden extreme multi-stability and pseudo-random sequence generator

A new memristor-based fractional-order chaotic system

In this paper, a new four-dimensional incommensurate fractional-order system is proposed by introducing an ideal flux-controlled memristor into a three-dimensional chaotic system, and combining it with fractional-order calculus theory, which is solved by using the Adomian decomposition method (ADM). Through theoretical analysis we found the system has numerous equilibrium points. Compared with the original system, the modified system exhibits richer dynamical behaviors. The main manifestations are: (i) Antimonotonicity varying with the initial value. (ii) Three kinds of transient transition behaviors: transient asymptotically-period (A-period), transient chaos, and tri-state transition (chaos-A-period-chaos). (iii) Initial offset boosting behavior. (iv) Hidden extreme multistability. (v) As the order q changes, the system is capable of generating a variety of asymptotically periodic attractors and chaotic attractors. These behaviors above are analyzed in detail by means of numerical simulations such as phase diagram, bifurcation diagram, Lyapunov exponent spectrum (LEs), time-series diagram, and attraction basin. Finally, the system is implemented with a hardware circuit based on a digital signal processor (DSP), which in turn proved the correctness of the numerical analysis simulations and the physical realizability of the system.

Study on a four-dimensional fractional-order system with dissipative and conservative properties

In this paper, based on an existing four-dimensional integer-order conservative system, the corresponding fractional-order system is proposed and solved by the Adomian decomposition method (ADM). In the process of analyzing the influence of parameters and initial values on the dynamic behaviors of system, we find various quasi-periodic and chaotic flows with different topological structures as well as hidden extreme multistability. In addition, we observe two types of transient transition behavior. It is worth noting that a new phenomenon is found, that is, the dynamic behaviors of system change from dissipative to conservative with the increase of order. Phase diagram , Lyapunov exponent spectrum, bifurcation diagram and spectral entropy (SE) complexity algorithm are used to analyze the above dynamic behaviors. Finally, the hardware implementation of system is realized on the DSP platform, and experimental results are consistent with numerical simulation results. The research results show that the fractional-order system has high complexity and better engineering application value.

A New Four-Dimensional Non-Hamiltonian Conservative Hyperchaotic System

This paper presents a new four-dimensional non-Hamiltonian conservative hyperchaotic system. In the absence of equilibrium points in the system, the phase trajectories generated by the system have hidden features. The conservative features that vary with the parameter have been analyzed in detail by Lyapunov exponent spectrum, bifurcation diagram, the sum of Lyapunov exponents, and the fractional dimensions, and during the analysis, multiple quasi-periodic four-dimensional tori as well as hyperchaotic attractors have been observed. The Poincaré sections confirm these dynamic behaviors. Amidst the process of studying the dynamical behavior of the system with initial values, the hidden extreme multistability, and the initial offset boosting behavior, the results have been witnessed for the very first time in a conservative chaotic system. The phase diagram and attraction basin also confirm this assertion, while two complex transient transition behaviors have been observed. Moreover, through the introduction of a spectral entropy algorithm, the complexity analysis of the time sequences generated by the system have been performed and compared with the existing literature. The results show that the system has a high degree of complexity. The design and construction of the analog circuit of the system for simulation, the circuit experimental results are consistent with the numerical simulation, further verifying the physical realizability of the newly proposed system. This lays a good foundation for its practical application in engineering.

Hidden Extreme Multistability in a Novel No-Equilibrium Fractional-Order Chaotic System and Its Synchronization Control

Chaotic systems with hidden attractors have received widespread attention recently. In this paper, we construct a new four-dimensional fractional-order chaotic system by adding a further variable in a 3D chaotic system. This newly presented system has no equilibrium points, which reveals that the attractors produced by this system are all hidden. The intricate hidden dynamic properties are investigated by adopting nonlinear dynamical analysis tools such as phase diagrams, bifurcation diagrams, Lyapunov exponents, and chaos diagrams. In particular, the coexisting hidden attractors and extreme multistability are also observed in the system. Furthermore, the electronic circuit of this chaotic system is designed and the corresponding hardware circuit experiment is also achieved. Finally, based on the theory of fractional finite time stability theorem, a suitable finite time synchronization controller is designed. Numerical simulations are provided to verify the effectiveness of the proposed synchronization scheme.

Complex dynamics from a novel memristive 6D hyperchaotic autonomous system

A simple 5D hyperchaotic system recently introduced in the literature is modified by using a charge-controlled memristor model and striking behaviors are uncovered. The resulting system is a 6D hyperchaotic system, which generates hidden attractors with the unusual feature of having plan and line equilibrium under different parameter conditions. Its dynamical behaviors are characterized through bifurcation diagrams, Lyapunov exponents, phase portraits, Poincare sections and time series. Rich nonlinear dynamics such as limit cycles, quasi-periodicity, chaos, hyperchaos, bursting and hidden extreme multistability are found for appropriate sets of parameter values. The high complexity of the system is confirmed by its Kaplan–yorke dimension (greater than five). Additionally, an electronic circuit is designed to implement the novel system and PSpice simulation results are in good accordance with the numerical investigations. To the best of our knowledge, this system is the first with higher order presenting all those phenomena.

Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh–Nagumo circuit

Due to the introduction of ideal memristors, extreme multistability has been found in many autonomous memristive circuits. However, such extreme multistability has not yet been reported in a non-autonomous memristive circuit. To this end, this paper presents an improved non-autonomous memristive FitzHugh–Nagumo circuit that possesses a smooth hyperbolic tangent memductance nonlinearity, from which coexisting infinitely many attractors are obtained. By utilizing voltage–current circuit model, a three-dimensional non-autonomous dynamical model is established, based on which the initial-dependent dynamics is explored by numerical plots and extreme multistability is thereby exhibited. To confirm that the improved non-autonomous memristive circuit operates in hidden oscillating patterns, an accurate two-dimensional non-autonomous dimensionality reduction model with initial-related parameters is further built by using incremental integral transformation, upon which stability analysis and bifurcation behaviors are elaborated. Because the equilibrium state of the dimensionality reduction model is always a stable node-focus, hidden extreme multistability with coexisting infinitely many attractors is truly confirmed. Finally, PSIM circuit simulations validate the initial-related hidden dynamical behaviors.

A novel memristive 6D hyperchaotic autonomous system with hidden extreme multistability

Abstract This paper presents a 6D autonomous system, obtained by introducing a flux-controlled memristor model into an existing 5D hyperchaotic autonomous system. The dynamics of the novel system are investigated in terms of equilibrium points, bifurcation diagrams, Lyapunov exponent spectra, phase portraits, time series and attraction basins. Moreover, its circuit implementation is also proposed. The system is hyperchaotic under a line or a plan of equilibria. Other attractive dynamics observed are: hidden extreme multistability, transient chaos, bursting and offset boosting phenomenon. Interestingly, for a particular set of parameters, an unusual metastability showing transition from chaotic to periodic bursting dynamics is found. Dynamical systems capable of producing the above complexity are not common in the literature; thus, this novel memristive 6D hyperchaotic autonomous system deserves particular attention. PSpice based simulations are in good agreement with numerical computations. To the best of our knowledge, this system is the first with higher order presenting all those phenomena.

Hidden extreme multistability in a novel 4D fractional-order chaotic system

The hidden attractor and extreme multistability are very important topics in nonlinear dynamics. In this paper, by adding a further variable in a three-dimensional chaotic system, a novel four-dimensional fractional-order chaotic system is developed. This system consists of eight terms including three different nonlinear terms and one constant term. What interests us is that this newly presented system has no-equilibrium but it can also exhibit rich and complex hidden dynamics. Furthermore, the offset boosting of a variable of the proposed chaotic system can be achieved by adjusting the constant term. The intricate hidden dynamic properties of the proposed chaotic system are investigated by employing conventional nonlinear dynamical analysis tools including equilibrium, phase planes, bifurcation diagrams and Lyapunov exponents, chaos diagrams, etc. Finally, Multisim simulations and the corresponding hardware experiments are implemented to validate the theoretical analysis.

Hidden Extreme Multistability, Antimonotonicity and Offset Boosting Control in a Novel Fractional-Order Hyperchaotic System Without Equilibrium

Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil’nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system.