Multistable Systems Research Articles

Chaos-Based Application of a Novel Multistable 5D Memristive Hyperchaotic System with Coexisting Multiple Attractors

Novel memristive hyperchaotic system designs and their engineering applications have received considerable critical attention. In this paper, a novel multistable 5D memristive hyperchaotic system and its application are introduced. The interesting aspect of this chaotic system is that it has different types of coexisting attractors, chaos, hyperchaos, periods, and limit cycles. First, a novel 5D memristive hyperchaotic system is proposed by introducing a flux-controlled memristor with quadratic nonlinearity into an existing 4D four-wing chaotic system as a feedback term. Then, the phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy are used to analyze the basic dynamics of the 5D memristive hyperchaotic system. For a specific set of parameters, we find an unusual metastability, which shows the transition from chaotic to periodic (period-2 and period-3) dynamics. Moreover, its circuit implementation is also proposed. By using the chaoticity of the novel hyperchaotic system, we have developed a random number generator (RNG) for practical image encryption applications. Furthermore, security analyses are carried out with the RNG and image encryption designs.

A New 4-D Multi-Stable Hyperchaotic Two-Scroll System with No-Equilibrium and its Hyperchaos Synchronization

A new 4-D multi-stable hyperchaotic two-scroll system with four quadratic nonlinearities is proposed in this paper. The dynamical properties of the new hyperchaotic system are described in terms of finding equilibrium points, phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. We discover that the new hyperchaotic system has no equilibrium point and hence it exhibits a hidden attractor. Furthermore, we show that the new hyperchaos system has multi-stability by the coexistence of hyperchaotic attractors for different values of initial conditions. As a control application, we use integral sliding mode control (ISMC) to derive new results for the hyperchaos synchronization of the new 4-D multi-stable hyperchaotic two-scroll system with hidden attractor.

Dynamics, control and symmetry-breaking aspects of a new chaotic Jerk system and its circuit implementation

The study of chaotic systems with symmetry is very well documented. However, very little is known unfortunately about the dynamics of such systems when their symmetry is perturbed or broken. This paper introduces a new chaotic jerk system with quadratic-cubic nonlinearity $$\varphi _{k} \left( x \right) =-x+kx^{2}+x^{3}$$ where k denotes a symmetry perturbation parameter. We consider the mechanism of chaos generation both for the symmetric ($$k=0$$) and the asymmetric ($$k\ne 0$$) modes of operation. The model has three fixed points one of which is located at the origin of the state space. Oscillations are excited from the origin, giving rise to a mono-scroll chaotic attractor. We demonstrate that the presence of the quadratic terms breaks the odd symmetry of the model and engenders a plethora of new nonlinear dynamics patterns such as critical transitions, coexisting asymmetric bubbles of bifurcation, coexisting asymmetric attractors, and parallel bifurcation branches. These interesting features are rarely reported, at least in a 3D autonomous system having the simplicity of the one studied in this work. By exploiting the method of linear augmentation, a simple control strategy is developed that help to convert the multistable system from the state of six coexisting attractors to a monostable state when simply adjusting the coupling parameter. Experimental results based on an appropriate electronic realization of the system are consistent with the theoretical investigations.

Tailored morphologies in two-dimensional ferronematic wells.

We focus on a dilute uniform suspension of magnetic nanoparticles in a nematic-filled micron-sized shallow well with tangent boundary conditions as a paradigm system with two coupled order parameters. This system exhibits spontaneous magnetization without magnetic fields. We numerically obtain the stable nematic and associated magnetization morphologies, induced purely by the geometry, the boundary conditions, and the coupling between the magnetic nanoparticles and the host nematic medium. Our most striking observations pertain to domain walls in the magnetization profile, whose location can be manipulated by the coupling and material properties, and stable interior and boundary nematic defects, whose location and multiplicity can be tailored by the coupling too. These tailored morphologies are not accessible in uncoupled systems and can be used for multistable systems with singularities and stable interfaces.

Thermodynamics of switching in multistable non-equilibrium systems.

Multistable non-equilibrium systems are abundant outcomes of nonlinear dynamics with feedback, but still relatively little is known about what determines the stability of the steady states and their switching rates in terms of entropy and entropy production. Here, we will link fluctuation theorems for the entropy production along trajectories with the action obtainable from the Freidlin-Wentzell theorem to elucidate the thermodynamics of switching between states in the large volume limit of multistable systems. We find that the entropy production at steady state plays no role, but the entropy production during switching is key. Steady-state entropy and diffusive noise strength can be neglected in this limit. The relevance to biological, ecological, and climate models is apparent.

Uncertain Destination of a 4D Autonomous System with Five Line Equilibria

Objectives: This paper introduces a novel 4D autonomous dynamic system with five line equilibria and a smooth nonlinearity. Methods/ statistical analysis: The new model is obtained by adding one more freedom degree to the 3D jerk system recently introduced by Kengne and Mogue, 2018. To analyze and study the model, Ruth criterion principle is used for the stability of different lines equilibria. Using traditional dynamics tools such as bifurcation diagrams, phase portraits, Poincare section, power spectrum, and Pspice software, the dynamic of the system is carried out. Findings: The new elegant system has an extremely rich dynamics predominated by the phenomenon of extreme multistability. The various sequences of coexisting route to chaos (coexisting bifurcation) confirm the uncertain destination of our novel elegant system. Note that, for the best of author’s knowledge, this is one of the best reproducible extreme multistable system because is not a flux control memristor-based system. Application/improvements: The results obtained in this investigation enrich the literature and being used to improve the extreme multistability application in many research domains such as Random Number Generation (RNG) and image encryption. Keywords: Five line equilibria, Extreme multistability, Composite tanh-cubic nonlinearity, PSpice simulations.

Secure Communication Scheme Based on a New 5D Multistable Four-Wing Memristive Hyperchaotic System with Disturbance Inputs

By introducing a flux-controlled memristor model with absolute value function, a 5D multistable four-wing memristive hyperchaotic system (FWMHS) with linear equilibrium points is proposed in this paper. The dynamic characteristics of the system are studied in terms of equilibrium point, perpetual point, bifurcation diagram, Lyapunov exponential spectrum, phase portraits, and spectral entropy. This system is of the group of systems that have coexisting attractors. In addition, the circuit implementation scheme is also proposed. Then, a secure communication scheme based on the proposed 5D multistable FWMHS with disturbance inputs is designed. Based on parametric modulation theory and Lyapunov stability theory, synchronization and secure communication between the transmitter and receiver are realized and two message signals are recovered by a convenient robust high-order sliding mode adaptive controller. Through the proposed adaptive controller, the unknown parameters can be identified accurately, the gain of the receiver system can be adjusted continuously, and the disturbance inputs of the transmitter and receiver can be suppressed effectively. Thereafter, the convergence of the proposed scheme is proven by means of an appropriate Lyapunov functional and the effectiveness of the theoretical results is testified via numerical simulations.

A 3-D Multi-Stable System With a Peanut-Shaped Equilibrium Curve: Circuit Design, FPGA Realization, and an Application to Image Encryption

A new 3-D chaotic dynamical system with a peanut-shaped closed curve of equilibrium points is introduced in this work. Since the new chaotic system has infinite number of rest points, the new chaotic model exhibits hidden attractors. A detailed dynamic analysis of the new chaotic model using bifurcation diagrams and entropy analysis is described. The new nonlinear plant shows multi-stability and coexisting convergent attractors. A circuit model using MultiSim of the new 3-D chaotic model is designed for engineering applications. The new multi-stable chaotic system is simulated on a field-programmable gate array (FPGA) by applying two numerical methods, showing results in good agreement with numerical simulations. Consequently, we utilize the properties of our chaotic system in designing a new cipher colour image mechanism. Experimental results demonstrate the efficiency of the presented encryption mechanism, whose outcomes suggest promising applications for our chaotic system in various cryptographic applications.

FPGA design and circuit implementation of a new four-dimensional multistable hyperchaotic system with coexisting attractors

This research work focuses upon the FPGA design and electronic circuit implementation of a new multistable four-dimensional hyperchaotic system. This work starts with the dynamics and phase plots of a new four-dimensional hyperchaos system. A detailed bifurcation analysis is carried out for the new hyperchaos system, and special properties such as multistability with coexisting attractors are reported for the new system. An electronic circuit model using MultiSim of the new hyperchaos system is designed. Finally, an FPGA-based design of the new hyperchaos system is performed by applying two numerical methods. Circuit simulation and FPGA design of the new hyperchaos system are very useful for practical applications of the new hyperchaos system.

A new multistable hyperjerk dynamical system with self-excited chaotic attractor, its complete synchronisation via backstepping control, circuit simulation and FPGA implementation

A new multistable hyperjerk dynamical system with self-excited chaotic attractor, its complete synchronisation via backstepping control, circuit simulation and FPGA implementation

A new multistable hyperjerk dynamical system with self-excited chaotic attractor, its complete synchronisation via backstepping control, circuit simulation and FPGA implementation

In this work, we report a new 4D chaotic hyperjerk system and present a detailed dynamic analysis of the new system with Lyapunov exponents, bifurcation plots, etc. We find that the new hyperjerk system exhibits multistability and coexisting chaotic attractors. The hyperjerk system has a unique saddle-focus rest point at the origin, which is unstable. This shows that the new chaos hyperjerk system has a self-excited chaotic attractor. As an application of backstepping control, we obtain new results for the global chaos complete synchronisation of pair of chaotic hyperjerk systems. A circuit model using MultiSim of the new chaotic hyperjerk system is designed for applications in practice. Finally, an FPGA-based implementation of the new chaotic hyperjerk dynamical system is performed by applying two numerical methods and their corresponding hardware resources are given.

FPGA design and circuit implementation of a new four-dimensional multistable hyperchaotic system with coexisting attractors

FPGA design and circuit implementation of a new four-dimensional multistable hyperchaotic system with coexisting attractors

A Complete Investigation of the Effect of External Force on a 3D Megastable Oscillator

Multistability is an essential topic in nonlinear dynamics. Recently, two critical subsets of multistable systems have been introduced: systems with extreme multistability and systems with megastability. In this paper, based on a newly introduced megastable system, a megastable forced oscillator is introduced. The effect of adding a forcing term and its parameters on the dynamical behavior of the designed system is investigated. By the help of bifurcation diagram and Lyapunov exponents, it is shown that the modified oscillator can show a variety of dynamical solutions including limit cycle, torus, and strange attractor.

Mitigation of tipping point transitions by time-delay feedback control.

In stochastic multistable systems driven by the gradient of a potential, transitions between equilibria are possible because of noise. We study the ability of linear delay feedback control to mitigate these transitions, ensuring that the system stays near a desirable equilibrium. For small delays, we show that the control term has two effects: (i) a stabilizing effect by deepening the potential well around the desirable equilibrium and (ii) a destabilizing effect by intensifying the noise by a factor of (1-τα)-1/2, where τ and α denote the delay and the control gain, respectively. As a result, successful mitigation depends on the competition between these two factors. We also derive analytical results that elucidate the choice of the appropriate control gain and delay that ensure successful mitigations. These results eliminate the need for any Monte Carlo simulations of the stochastic differential equations and, therefore, significantly reduce the computational cost of determining the suitable control parameters. We demonstrate the application of our results on two examples.

Noise-induced resistive switching in a memristor based on ZrO2(Y)/Ta2O5 stack

Resistive switching (RS) is studied in a memristor based on a ZrO2(Y)/Ta2O5 stack under a white Gaussian noise voltage signal. We have found that the memristor switches between the low resistance state and the high resistance state in a random telegraphic signal (RTS) mode. The effective potential profile of the memristor shows from two to three local minima and depends on the input noise parameters and the memristor operation. These observations indicate the multiplicative character of the noise on the dynamical behavior of the memristor, that is the noise perceived by the memristor depends on the state of the system and its electrical properties are influenced by the noise signal. The detected effects manifest the fundamental intrinsic properties of the memristor as a multistable nonlinear system.

A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: Dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method

Four-dimensional chaotic systems are a very interesting topic for researchers, given their special features. This paper presents a novel fractional-order four-dimensional chaotic system with self-excited and hidden attractors, which includes only one constant term. The proposed system presents the phenomenon of multi-stability, which means that two or more different dynamics are generated from different initial conditions. It is one of few published works in the last five years belonging to the aforementioned category. Using Lyapunov exponents, the chaotic behavior of the dynamical system is characterized, and the sensitivity of the system to initial conditions is determined. Also, systematic studies of the hidden chaotic behavior in the proposed system are performed using phase portraits and bifurcation transition diagrams. Moreover, a design technique of a new fuzzy adaptive sliding mode control (FASMC) for synchronization of the fractional-order systems has been offered. This control technique combines an adaptive regulation scheme and a fuzzy logic controller with conventional sliding mode control for the synchronization of fractional-order systems. Applying Lyapunov stability theorem, the proposed control technique ensures that the master and slave chaotic systems are synchronized in the presence of dynamic uncertainties and external disturbances. The proposed control technique not only provides high performance in the presence of the dynamic uncertainties and external disturbances, but also avoids the phenomenon of chattering. Simulation results have been presented to illustrate the effectiveness of the presented control scheme.

Dynamics and high-efficiency of a novel multi-stable energy harvesting system

Scavenging energy from external vibration with low-frequencies and low-level intensities has always been a huge challenge for the energy harvesting. To remedy this key issue, this study is to present a novel multi-stable electromagnetic transduction energy harvesting system by adjusting the geometric parameters of linear springs. Here, we prove that the multi-stable energy harvesting system composed of a nonlinear restoring mechanism can significantly improve the efficiency of the energy converter. The electromechanical model is set up, and the corresponding equations are derived by the Extended Averaging Method and Fokker–Planck equation analysis. The response under harmonic excitation exhibits that, compared to a mono-stable and a bi-stable system, with a tri-stable or a quad-stable system, we can achieve a higher-energy interwell motion with large-amplitude periodic oscillation, leading to significant increases of the harvested current and power. Moreover, improved the expected value of the mean square current and mean harvested power can also be attained under the Gaussian white noise with a small intensity. The results show that the proposed multi-stable system is a feasible design that can improve the power capture performance of energy harvesting from low-level excitations.

On some methods for generating extremely multistable systems

A multistable dynamic system can demonstrate solutions with fundamentally different behavior depending on the choice of their initial conditions, which poses a threat to its use in practical engineering applications. On the other hand, the system’s multistability may be its undisputed advantage when it is used, for example, to hide information in communication systems and audio encryption schemes to improve the performance of secure communications, since in this case a special choice of initial conditions can play the role of a “secret key”. In this paper, we propose some approaches to the generation of extremely multistable systems containing an infinite number of attractors using mathematical models of systems in Lurie form.

Switching thresholds for multistable systems under strong external perturbation

Multistability is a common feature of various dynamical systems which manifests itself as the possibility to demonstrate various behaviors for the same parameter values. These behaviors or states of the system must be stable against weak perturbation in order to be observable in real life. However, a strong enough perturbation may destroy a certain state and lead to the system switching to another one. From the viewpoint of nonlinear dynamics, the response of a multistable system to a strong stimulus depends on the configuration and the mutual arrangement of basins of different attractors. In the present paper, we introduce a novel measure for characterization of a multistable system, the switching threshold. It equals the amplitude of a minimal perturbation capable of switching the system from one attractor to another. We develop a numerical algorithm for calculation of the switching thresholds and apply it to a number of paradigmatic models including dynamical networks. We show that the values of switching thresholds provide important information about multistable systems and their responses to external stimuli. This information allows to develop methods of optimal control of multistable systems by external signals. Surprisingly, it also allows to predict some features of their dynamics under the influence of external noise.

Generation of Extremely Multistable Systems Based on Lurie Systems

Chaotic signals and systems are widely used in image encryption, secure communications, weak signal detection, and radar systems. Many researchers have focused in recent years on the development of systems with an infinite number of coexisting chaotic attractors. We propose some approaches in this work to the generation of self-reproducing systems with an infinite number of coexisting self-excited or hidden chaotic attractors with the same Lyapunov exponents based on mathematical models of Lurie systems. The proposed approach makes it possible to generate extremely multistable systems using numerous known examples of the existence of chaotic attractors in Lurie systems without resorting to an exhaustive computer search. We illustrate the proposed methods by constructing extremely multistable systems with 1-D and 2-D grids of hidden chaotic attractors using the generalized Chua system, in which hidden attractors were first discovered by G.A. Leonov and N.V. Kuznetsov.