Multistability Phenomena Research Articles

Multistability phenomenon in signal processing, energy harvesting, composite structures, and metamaterials: A review

Multistability is the phenomenon of multiple coexistent stable states, which are highly sensitive to perturbations, initial conditions, system parameters, etc. Multistability has been widely found in various scientific areas including biology, physics, chemistry, climatology, sociology, and ecology. In a number of systems where multistability naturally exists, it is found to be undesirable because of the involuntary interwell or chaotic switching among dynamical states that disorder the systems and cause instability. However, in recent decades, researchers have identified numerous benefits of multistability and have devoted research efforts to artificially creating it for a wide range of applications, including signal processing, energy harvesting, composite structures and metamaterials, and micro-/nano-electromechanical actuators. This is because of the unique characteristics of multistability, such as rich potential structure, interwell dynamics, broadband nature, and alleviation of the input energy to sustain stable states, which may play different advantageous roles depending on their applications. In this review, we introduce how researchers create the key of multistability and utilize it to open a new world of theories, materials, and structures. We concentrate on developing histories from bistability to multistability in several potential applications. Different designs of digital and physical multistable systems, and their modeling, performance quantifiers, advantageous mechanisms, and improved techniques are reviewed and discussed in depth. Furthermore, we summarize the key issues and challenges of application-oriented multistability and the corresponding possible solutions, from the phenomenon itself to its realistic implementation. Finally, we provide the prospects for future studies on multistability in more developing research fields.

Extreme Multistability and Antimonotonicity in a Shinriki Oscillator with Two Flux-Controlled Memristors

This paper proposes a novel memristive chaotic circuit which originated from a Shinriki oscillator with two flux-controlled memristors of different polarities. This two-memristor-based Shinriki oscillator (TMSO) having a special plane equilibrium is prone to exhibiting the initial-dependent phenomenon of extreme multistability. To investigate its internal dynamics, a third-order dimensionality reduction model is established by utilizing the constitutive relationship of its memristor’s flux and charge. The uncertain plane equilibrium is transfered into some deterministic model that can accurately predict the dynamical evolution of the system, where interesting phenomena of asymmetric bifurcations, extreme multistability and antimonotonicity are detected and analyzed by evaluating the position and stability of the equilibria in the flux–charge model. The simulation is carried out via Multisim to validate the analysis model, and the comparison of the phase trajectories, before and after dimensionality reduction, shows that this oscillator is good for research and practical use.

Optical bistable and multistable phenomena in aperiodic multilayer structures with graphene

We investigate the optical bistability, tristability and multistability in a complex structure which composed by graphene and dielectric multilayers submitted to the Thue-Morse (TM) sequence. Dielectric multilayers pattern spatially to meet the TM sequence rules and graphene is embedded at the interfaces of dielectrics. The TM lattices support discrete resonant modes and strong field localization which can greatly enhance the nonlinear effect of graphene, and sequentially render low-threshold double bistability. The thresholds of bistability and the intervals between the upper thresholds and lower thresholds increase as the chemical potential of graphene is increasing. Furthermore, tristability and multistability are also induced by modulating the incident wavelength with a large red-detuning. This study paves a new way towards all-optical multivalues switches, optical logic elements and optical storage. • The strong field localization in Thue-Morse sequence greatly enhance the nonlinearity of graphene. • Low threshold double optical bistability and multistability are induced around the defect modes. • The threshold of bistability and multistability can be changed by the chemical potential of graphene flexibly.

Dynamical analysis of series hybrid electric vehicle powertrain with torsional vibration: Antimonotonicity and coexisting attractors

In this paper, nonlinear dynamics of the torsional vibrations of the Series Hybrid Electric Vehicle (SHEV) powertrain are explored. The mathematical equation of the SHEV is obtained by considering both internal and external excitations. We examine the effects of rotational speed, load fluctuation, and road fluctuation on the dynamics of the proposed model. Preliminary analysis based on Hamiltonian energy shows that there is sufficient energy to maintain continuous oscillation behaviors. Our study shows that the interaction between internal and external perturbations originates an important set of phenomena, including chaos, when varying system parameters. More interestingly, the exciting phenomenon of multistability is investigated for the first time on the proposed SHEV model. Furthermore, when increasing excitation force amplitude in some specific regions, an interesting scenario of antimonotonicity is found. • Nonlinear dynamics of the torsional vibrations of the Series Hybrid Electric Vehicle (SHEV) powertrain are explored. • Both internal and external excitations are considered in the mathematical model. • The Hamiltonian energy shows that there is sufficient energy to maintain continuous oscillation behaviors. • The exciting phenomenon of multi-stability is investigated on the proposed SHEV model. • Anti-monotonicity and period-doubling bifurcations are also reported when adjusting the excitation force amplitude of the system.

Symmetric coexisting attractors and extreme multistability in chaotic system

In this paper, a new four-dimensional (4D) chaotic system with two cubic nonlinear terms is proposed. The most striking feature is that the new system can exhibit completely symmetric coexisting bifurcation behaviors and four symmetric coexisting attractors with the same Lyapunov exponents in all parameter ranges of the system when taking different initial states. Interestingly, these symmetric coexisting attractors can be considered as unusual symmetrical rotational coexisting attractors, which is a very fascinating phenomenon. Furthermore, by using a memristor to replace the coupling resistor of the new system, an interesting 4D memristive hyperchaotic system with one unstable origin is constructed. Of particular surprise is that it can exhibit four groups of extreme multistability phenomenon of infinitely many coexisting attractors of symmetric distribution about the origin. By using phase portraits, Lyapunov exponent spectra, and coexisting bifurcation diagrams, the dynamics of the two systems are fully analyzed and investigated. Finally, the electronic circuit model of the new system is designed for verifying the feasibility of the new chaotic system.

Recent Advances in Dimensionality Reduction Modeling and Multistability Reconstitution of Memristive Circuit

Due to the introduction of memristors, the memristor-based nonlinear oscillator circuits readily present the state initial-dependent multistability (or extreme multistability), i.e., coexisting multiple attractors (or coexisting infinitely many attractors). The dimensionality reduction modeling for a memristive circuit is carried out to realize accurate prediction, quantitative analysis, and physical control of its multistability, which has become one of the hottest research topics in the field of information science. Based on these considerations, this paper briefly reviews the specific multistability phenomenon generating from the memristive circuit in the voltage-current domain and expounds the multistability control strategy. Then, this paper introduces the accurate flux-charge constitutive relation of memristors. Afterwards, the dimensionality reduction modeling method of the memristive circuits, i.e., the incremental flux-charge analysis method, is emphatically introduced, whose core idea is to implement the explicit expressions of the initial conditions in the flux-charge model and to discuss the feasibility and effectiveness of the multistability reconstitution of the memristive circuits using their flux-charge models. Furthermore, the incremental integral transformation method for modeling of the memristive system is reviewed by following the idea of the incremental flux-charge analysis method. The theory and application promotion of the dimensionality reduction modeling and multistability reconstitution are proceeded, and the application prospect is prospected by taking the synchronization application of the memristor-coupled system as an example.

Complex Behavior of a Hyperchaotic TNC Oscillator: Coexisting Bursting, Space Magnetization, Control of Multistability and Application in Image Encryption Based on DNA Coding

In this work, several new aspects of the dynamics of the well-known TNC hyperchaotic oscillator are investigated. Numerous novelties appear in this work, namely particular structures of space magnetization, the coexistence of bursting patterns, the coexistence of up to four asymmetric different attractors, offset-boosting, and antimonotonicity. In addition to this interesting and particular combination of features in the TNC oscillator, the control of multistability phenomenon is carried out using linear augmentation control scheme. Finally, knowing that the protection of digital images is of particularly great interest, the complexity of pseudo-random hyperchaotic sequences of the TNC oscillator is combined with the similarity that the DNA coding shares with the binary code to build a new image encryption algorithm with strong robustness and high speed. This algorithm is successfully evaluated using cropping attack, noise attack and differential attack. The results obtained demonstrate that the proposed algorithm is efficient and of good quality.

A Novel Memristor Chaotic System with a Hidden Attractor and Multistability and Its Implementation in a Circuit

By introducing an ideal and active flux-controlled memristor and tangent function into an existing chaotic system, an interesting memristor-based self-replication chaotic system is proposed. The most striking feature is that this system has infinite line equilibria and exhibits the extreme multistability phenomenon of coexisting infinitely many attractors. In this paper, bifurcation diagrams and Lyapunov exponential spectrum are used to analyze in detail the influence of various parameter changes on the dynamic behavior of the system; it shows that the newly proposed chaotic system has the phenomenon of alternating chaos and limit cycle. Especially, transition behavior of the transient period with steady chaos can be also found for some initial conditions. Moreover, a hardware circuit is designed by PSpice and fabricated, and its experimental results effectively verify the truth of extreme multistability.

New hyperchaotic system with single nonlinearity, its electronic circuit and encryption design based on current conveyor

Nowadays, hyperchaotic system (HCSs) have been started to be used in engineering applications because they have complex dynamics, randomness, and high sensitivity. For this purpose, HCSs with different features have been introduced in the literature. In this work, a new HCS with a single discontinuous nonlinearity is introduced and analyzed. The proposed system has one saddle focus equilibrium. When the dynamic properties and bifurcation graphics of the system are analyzed, it is determined that the proposed system exhibits the complex phenomenon of multistability. Moreover, analog electronic circuit design of the proposed system is performed with positive second-generation current conveyor. In addition, an encryption circuit is designed to demonstrate that the proposed system can be used in various engineering applications.

Blinking Networks of Memristor Oscillatory Circuits in the Flux-Charge Domain.

Multistability phenomena and complex nonlinear dynamics in memristor oscillators pave the way to obtain efficient solutions to optimization problems by means of novel computational architectures based on the interconnection of single–device oscillators. It is well-known that topological properties of interconnections permit to control synchronization and spatio–temporal patterns in oscillatory networks. When the interconnections can change in time with a given probability to connect two oscillators, the whole network acts as a complex network with blinking couplings. The work of has shown that a particular class of blinking complex networks are able to completely synchronize in a faster fashion with respect to other coupling strategies. This work focuses on the specific class of blinking complex networks made of Memristor–based Oscillatory Circuits (MOCs). By exploiting the recent Flux–Charge Analysis Method, we make clear that synchronization phenomena in blinking networks of memristor oscillators having stochastic couplings, i.e., Blinking Memristor Oscillatory Networks (BMONs), correspond to global periodic oscillations on invariant manifolds and the effect of a blinking link is to shift the nonlinear dynamics through the infinite (invariant) manifolds. Numerical simulations performed on MOCs prove that synchronization phenomena can be controlled just by changing the coupling amongst them.

A novel memristive chaotic system without any equilibrium point

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.

Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator

A predator–prey model with functional response Holling type II, Allee effect in the prey and a generalist predator is considered. It is shown that the model with strong Allee effect has at most two positive equilibrium points in the first quadrant, one is always a saddle point and the other exhibits multi-stability phenomenon since the equilibrium point can be stable or unstable. The model with weak Allee effect has at most three positive equilibrium points in the first quadrant, one is always a saddle point and the other two can be stable or unstable node. In addition, when the parameters vary in a small neighbourhood of system parameters the model undergoes different bifurcations, such as saddle–node, Hopf and Bogdanov–Takens bifurcations. Moreover, numerical simulation is used to illustrate the impact in the stability of positive equilibrium point(s) by adding an Allee effect and an alternative food sources for predators.

A novel no-equilibrium HR neuron model with hidden homogeneous extreme multistability

• A novel no-equilibrium HR neuron model with memristive electromagnetic induction is proposed. • This memristive HR neuron model can exhibit complicated memristor initial offset boosting dynamics. • The interesting phenomenon of hidden homogeneous extreme multistability is found. • Hardware experiments are performed to verify homogeneous coexisting hidden attractors. • A pseudo-random number generator with excellent randomness is designed based on memristor initial-controlled chaotic sequences. In this paper, a novel no-equilibrium Hindmarsh-Rose (HR) neuron model with memristive electromagnetic induction is proposed. This memristive HR neuron model exhibits complex memristor initial offset boosting dynamics, from which infinitely many coexisting hidden attractors sharing the same shape but with different positions can be generated, therefore breeding the interesting phenomenon of hidden homogeneous extreme multistability. The complicated dynamical behaviors are detailedly investigated via bifurcation diagrams, Lyapunov exponents , time series, attraction basins and spectral entropy (SE) complexity. Moreover, PSIM circuit simulations and DSP hardware experiments are carried out to demonstrate the theoretical analyses and numerical simulations. Finally, a pseudorandom number generator is also designed by using the memristor initial-controlled chaotic sequences extracted from the memristive HR neuron model. The performance analysis results show that these chaotic sequences can yield pseudorandom numbers with excellent randomness, which are more suitable for chaos-based engineering applications.

No-argument memristive hyper-jerk system and its coexisting chaotic bubbles boosted by initial conditions

Abstract Antimonotonicity is used to characterize the specific dynamics for the generation of periodic orbits cascaded by their destruction, which is generally involved with the change of a bifurcation parameter. However, this phenomenon of antimonotonicity is rarely induced by the initial conditions of chaotic systems. To this end, this paper presents a no-argument memristive hyper-jerk system. When taking the initial condition of the memristor inner state (called as “memristor initial condition” for short) as a bifurcation parameter, we disclose the chaotic bubbles located in the primary interval and thereby display the coexisting infinitely many attractors with extreme multi-stability. Afterwards, when switching the memristor initial condition, we also uncover the initial condition-boosted coexisting chaotic bubbles theoretically and numerically. Therefore, the complex phenomenon of the extreme multi-stability with the initial condition-boosted coexisting chaotic bubbles is well revealed. Furthermore, a reconstituted model with the initial conditions in an explicit form is established in integral domain and the extreme multi-stability can be effectively interpreted through the stability analysis of the determined equilibrium points of the reconstituted model. Finally, a digital hardware platform is implemented to verify the memristor initial condition-dependent chaotic bubbles.

Spectral Entropy Analysis and Synchronization of a Multi-Stable Fractional-Order Chaotic System using a Novel Neural Network-Based Chattering-Free Sliding Mode Technique

Abstract An immense body of research has focused on chaotic systems, mainly because of their interesting applications in a wide variety of fields. A comprehensive understanding and synchronization of chaotic systems play pivotal roles in practical applications. To this end, the present study investigates a multi-stable fractional-order chaotic system. Firstly, some dynamical features of the system are described, and the chaotic behaviour of the system is verified. Then, both spectral entropy and spectral Min-Entropy are computed, and the phenomenon of multi-stability is shown. Besides, the combination of a new chattering-free robust sliding mode controller with a neural network observer is proposed for the synchronization of the fractional-order system. With the neural network estimator, unknown functions of the system are obtained, and the effects of disturbances are completely taken into account. Also, based on the Lyapunov stability theorem, the asymptotical stability of the closed-loop system is confirmed. Lastly, the proposed control technique is applied to the fractional-order system. Numerical results demonstrate the chattering-free and effective performance of the proposed control method for uncertain systems in the presence of unknown time-varying external disturbances.

Analysis of the ROA of an anaerobic digestion process via data-driven Koopman operator

Abstract Nonlinear biochemical systems such as the anaerobic digestion process experience the problem of the multi-stability phenomena, and thus, the dynamic spectrum of the system has several undesired equilibrium states. As a result, the selection of initial conditions and operating parameters to avoid such states is of importance. In this work, we present a data-driven approach, which relies on the generation of several system trajectories of the anaerobic digestion system and the construction of a data-driven Koopman operator to give a concise criterion for the classification of arbitrary initial conditions in the state space. Unlike other approximation methods, the criterion does not rely on difficult geometrical analysis of the identified boundaries to produce the classification.

Asymmetric coexisting bifurcations and multi-stability in an asymmetric memristive diode-bridge-based Jerk circuit

An asymmetric memristive diode-bridge (MDB) emulator is raised to imitate the asymmetric volt-ampere characteristic of a physical memristor. Then, an asymmetric MDB-based Jerk circuit is built and its state equation is derived, upon which the theoretical analysis, MATLAB-based numerical simulations, and hardware measurements are executed to reveal the asymmetric coexisting bifurcations and the phenomenon of multi-stability. The memristive Jerk circuit has three equilibrium points of a pair nontrivial equilibrium points of asymmetric unstable saddle-foci and a zero equilibrium point of unstable saddle-focus, which leads to the occurrence of asymmetric coexisting bifurcations and asymmetric local attraction basins. The asymmetrical bifurcations are numerically disclosed by 1-D/2-D bifurcation plots, Lyapunov spectrum, and phase plane trajectories. Multi-stability with asymmetric coexisting attractors under two sets of system parameters are demonstrated as examples by local attraction basins and phase plane trajectories. Thereafter, experimental circuit prototype employing discrete components is manually welded and hardware measurements are executed to validate the numerical simulations.

Multistability and chaotic scenario in a quantum pair-ion plasma

Abstract Multistability and chaotic scenario of arbitrary amplitude ion-acoustic waves in a quantum plasma consisting of negative ions, positive ions and electrons are investigated. The normalized basic equations are transformed to a four dimensional conservative dynamical system by introducing a travelling wave variable. Stability of the fixed points for the corresponding linearized system is briefly examined. Chaotic and quasi-periodic features of the arbitrary amplitude ion-acoustic waves are discussed using effective tools, viz. phase orientations, time series graph and graphs of Lyapunov exponents. Multistability phenomena is established with the help of phase spaces, largest Lyapunov exponents and cross-section of basins of attraction. The chaotic phenomena is further verified by 0−1 test. Results of this study can be applied in understanding dynamical phenomena of arbitrary amplitude ion-acoustic waves in quantum pair-ion plasmas.

Dynamic Analysis and Circuit Realization of a Novel No-Equilibrium 5D Memristive Hyperchaotic System with Hidden Extreme Multistability

In this paper, a novel no-equilibrium 5D memristive hyperchaotic system is proposed, which is achieved by introducing an ideal flux-controlled memristor model and two constant terms into an improved 4D self-excited hyperchaotic system. The system parameters-dependent and memristor initial conditions-dependent dynamical characteristics of the proposed memristive hyperchaotic system are investigated in terms of phase portrait, Lyapunov exponent spectrum, bifurcation diagram, Poincaré map, and time series. Then, the hidden dynamic attractors such as periodic, quasiperiodic, chaotic, and hyperchaotic attractors are found under the variation of its system parameters. Meanwhile, the most striking phenomena of hidden extreme multistability, transient hyperchaotic behavior, and offset boosting control are revealed for appropriate sets of the memristor and other initial conditions. Finally, a hardware electronic circuit is designed, and the experimental results are well consistent with the numerical simulations, which demonstrate the feasibility of this novel 5D memristive hyperchaotic system.

Suppression of self-oscillations and formation of heterogeneous structures by diffusion in the non-linear glycolytic model

A dynamical distributed model of glycolysis with the diffusion is considered in the parametric zone of self-oscillations. A phenomenon of the diffusion-induced suppression of self-oscillations is found and studied by technique of harmonic coefficients. We show how, under increase of diffusion, temporal oscillations of homogeneous solutions transform into stationary non-homogeneous structures in the form of patterns-attractors. A phenomenon of multistability in this spatially distributed glycolytic model is discussed and a variety of coexisting patterns and their amplitude characteristics is quantified.