Phenomenon Of Extreme Multistability Research Articles

Dynamical analysis and circuit implementation of a memristive chaotic system with infinite coexisting attractors

In order to obtain a chaotic system with more complex dynamic characteristics and more suitable for engineering applications, this paper combines a general memristor containing a hyperbolic tangent function with a simple three-dimensional chaotic system to construct a four-dimensional memristive chaotic system with infinite coexisting attractors. The memristive chaotic system is thoroughly studied through numerical simulations of various nonlinear systems, including the Lyapunov exponent spectra, bifurcation diagram, C0 complexity, two-parameter bifurcation diagram and basins of attraction. The analysis reveals that this system has complex dynamical behavior. It includes not only periodic limit loops and chaotic attractors that depend on the variation of system parameters, but also the extreme multi-stability phenomenon of infinite coexisting attractors that depend on the variation of the initial conditions of the system. In addition, the chaos degradation and offset boosting control of the system are also studied and analyzed. Finally, the correctness and realizability of the memristive chaotic system are verified by circuit simulation and hardware circuit fabrication.The experimental results show that this memristive chaotic system can lay the foundation for practical engineering fields such as secure communication and image encryption.

A novel fractional-order hyperchaotic complex system and its synchronization

A novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The results show abundant dynamical characteristics. Particularly, the phenomena of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by numerical simulations. It lays the foundation for practical applications of the proposed system.

Symmetric extreme multistability and memristor initial-offset boosting in a new 5D four-wing hyperchaotic system based on dual memristors

Based on a 5D two-wing hyperchaotic system with dual memristors, a new unified 5D four-wing hyperchaotic system based on dual memristors is proposed, which has four cases and can exhibit three different structures of four-wing attractors. This paper studies the dynamic behaviors of one of these cases by the phase diagrams, bifurcation diagrams, Lyapunov exponents, etc. Of particular surprise, this system has a space equilibrium and can exhibit the two groups of extreme multistability phenomenon of symmetric distribution about the origin. In addition, the striking and complex memristor initial boosting behaviors are investigated in this paper, which shows that the interesting phenomenon is nonlinear and different attractor structures with four-wing and two-wing. When a single parameter is changed, the initial boosting behavior has obvious changes, indicating the memristive initial conditions and parameters have an important influence on the position shift of the attractors. Finally, the new four-wing system is realized by Multisim circuit simulation, which verifies the feasibility of the new system.

Initial-Condition Effects on a Two-Memristor-Based Jerk System

Memristor-based systems can exhibit the phenomenon of extreme multi-stability, which results in the coexistence of infinitely many attractors. However, most of the recently published literature focuses on the extreme multi-stability related to memristor initial conditions rather than non-memristor initial conditions. In this paper, we present a new five-dimensional (5-D) two-memristor-based jerk (TMJ) system and study complex dynamical effects induced by memristor and non-memristor initial conditions therein. Using multiple numerical methods, coupling-coefficient-reliant dynamical behaviors under different memristor initial conditions are disclosed, and the dynamical effects of the memristor initial conditions under different non-memristor initial conditions are revealed. The numerical results show that the dynamical behaviors of the 5-D TMJ system are not only dependent on the coupling coefficients, but also dependent on the memristor and non-memristor initial conditions. In addition, with the analog and digital implementations of the 5-D TMJ system, PSIM circuit simulations and microcontroller-based hardware experiments validate the numerical results.

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.

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.

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.

Complex dynamics of a modified four order Wien-bridge oscillator model and FPGA implementation

This paper presents a novel fourth-order autonomous flux controlled memristive Wien-bridge system. Standard nonlinear diagnostic tools such as bifurcation diagram, graphs of largest Lyapunov exponent, Lyapunov stability diagram, phase space trajectory and isospike diagram are used to characterize dynamics of the system. Results show that the system presents hidden attractors with infinite equilibrium points known as line equilibrium for a suitable set of its parameters. The system also exhibits striking phenomenon of extreme multistability. Through isospike and Lyapunov stability diagram, spiral bifurcation leading to a center hub point is observed in a Wien-bridge circuit for the first time and the Field Programmable Gate Array -based implementation is performed to confirm its feasibility.

A Novel Non-Autonomous Chaotic System With Infinite 2-D Lattice of Attractors and Bursting Oscillations

In this brief, a novel three-dimensional non-autonomous chaotic system with periodic excitation and trigonometric function is proposed. Interestingly, with the disturbance of the periodic excitation, the system exhibits complex dynamical behaviors, including bursting oscillations (BOs), chaotic and hyperchaotic attractor. More importantly, because of the presence of trigonometric function, the system possesses infinite number of equilibrium points, which leads to the phenomenon of extreme multistability, namely infinite coexistence attractors and BOs. Besides, a variety of dynamic analysis tools such as phase diagram (PD), transformed phase diagram (TPD), time series (TS), bifurcation diagram (BD) and Lyapunov exponents (LE) are used to comprehensively analyze these interesting dynamics. Finally, an analog circuit is designed through the use of circuit simulation software PSPICE and realized by an experimental set-up to verify these dynamics.

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.

Extreme Multistability in Simple Area-Preserving Map

Initial condition-relied extreme multistability has been recently found in many continuous dynamical systems. However, such a specific phenomenon has not yet been discovered in a discrete iterative map. To investigate this phenomenon, this paper proposes a two-dimensional conservative map only with one sine nonlinearity. The proposed simple discrete map is area-preserving in the phase space and displays the coexistence of infinite chaotic and quasi-periodic orbits caused by infinite fixed points. Multiple numerical results indicate that the area-preserving chaotic and quasi-periodic orbits have the initial condition-relied quasi-periodic route to chaos and initial condition-boosting bifurcation dynamics, which allow the simple area-preserving map to emerge the complex phenomenon of extreme multistability. Furthermore, a microcontroller-based hardware platform is developed to implement the initial condition-boosting chaotic signals.

Flux–Charge Analysis of Two-Memristor-Based Chua's Circuit: Dimensionality Decreasing Model for Detecting Extreme Multistability

In this paper, from a new perspective of flux and charge, we present in-depth analyses of two ideal memristor emulators and the fifth-order memristive Chua's circuit constructed based on them. The constitutive flux–charge relations of the two adopted memristor emulators are first formulated, and their initial-dependent characteristics are numerically revealed and experimentally verified. Thereafter, with these two constitutive relations, a third-order dimensionality decreasing flux–charge model for the fifth-order memristive Chua's circuit is constructed, in which five extra constant system parameters are introduced to indicate the initial states of the five dynamic elements. Numerical simulations confirm that this newly constructed model possesses several determined equilibria and maintains the initial-dependent dynamics of the original voltage–current model. Thus, the complex and sensitive initial state-related extreme multistability phenomenon can be deeply explored through theoretical analyses and hardware measurements. It is demonstrated that the sensitive extreme multistability phenomenon becomes detectable in the flux–charge domain, which is efficient for exploring the inner mechanisms and further seeking possible applications of this special phenomenon.

Coexisting multiscroll hyperchaotic attractors generated from a novel memristive jerk system

In this paper, two kinds of novel non-ideal voltage-controlled multi-piecewise cubic nonlinearity memristors and their mathematical models are presented. By adding the memristor to the circuit of a three-dimensional jerk system, a novel memristive multiscroll hyperchaotic jerk system is established without introducing any other ordinary nonlinear functions, from which $$2N+2$$ -scroll and $$2M+1$$ -scroll hyperchaotic attractors are achieved. It is exciting to note that this new memristive system can produce the extreme multistability phenomenon of coexisting infinitely multiple attractors. Furthermore, the dynamical behaviours of the proposed system are analysed by phase portraits, equilibrium points, Lyapunov exponents and bifurcation diagrams. The results indicate that the system exhibits hyperchaotic, chaotic and periodic dynamics. Especially, the phenomenon of transient chaos can also be found in this memristive multiscroll system. Additionally, the MULTISIM circuit simulations and the hardware experimental results are performed to verify numerical simulations.

A simple inductor-free memristive circuit with three line equilibria

In this work, a novel inductor-free fourth-order two-memristor-based chaotic circuit is proposed. This new circuit is developed from a current feedback op amp-based sinusoidal oscillator through replacing a linear resistor with a memristor and adding another different parallel memristor to the cascaded memristor–capacitor net. The proposed circuit can perform chaotic, fixed point, and period behaviors. The most striking feature is that this system has three line equilibria and exhibits the extreme multistability phenomenon of the coexisting infinitely many attractors. Specially, amplitude death behavior and transient transition behavior can also be found in the proposed system. By using standard nonlinear analysis tools including system dissipation, equilibrium point stability, phase portrait, Lyapunov exponent spectrum, and bifurcation diagram, the fundamental dynamical characteristics of the circuit are investigated in detail. Moreover, a MULTISIM circuit is designed to verify the numerical simulations.

Uncertain destination dynamics of a novel memristive 4D autonomous system

This paper studies the dynamics of a novel memristive 4D autonomous system obtained by replacing the memristive diode bridge in the original circuit of [Chaos, Solitons and Fractals 91 180–197 (2016)] with a flux control memristor. The new memristor oscillator is described by a continuous time four-dimensional autonomous system with a line of equilibria. The analysis is carried out in terms of its parameters by using bifurcation diagrams, phase space trajectory plots, Poincaré sections, bifurcation like sequences, and graphs of Lyapunov exponents. The system is shown to experience the unusual phenomenon of extreme multistability characterized by the possibility of an infinite number of attractors for the same parameters setting. Period doubling and symmetry restoring crisis scenarios are reported. To the best of the authors’ knowledge, the results of this work represent the first report on the phenomenon of extreme multistability in a jerk system and thus deserve dissemination.

Extreme multi-stability in hyperjerk memristive system with hidden attractors and its adaptive synchronisation scheme

This paper presents a study of the phenomenon of extreme multi-stability in a novel 4D hyperjerk memristive system. The proposed system appertains to the category of dynamical systems with hidden attractors due to infinite equilibrium points. The behaviour of the system is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram and Lyapunov exponents. Also, this work showed that the extreme multi-stability phenomenon of the behaviour of infinitely many coexisting attractors depends on the initial conditions of the variables of the system. Moreover, the case of chaos synchronisation of the system with unknown parameters, using adaptive synchronisation method, is investigated.

Controlling extreme multistability of memristor emulator-based dynamical circuit in flux–charge domain

Memristive circuit with infinitely many equilibrium points can exhibit the special phenomenon of extreme multistability, whose dynamics mechanism and physical control are significant issues deserving in-depth investigations. In this paper, a control strategy for extreme multistability exhibited in an active band pass filter-based memristive circuit is explored in flux–charge domain. To this end, an incremental flux–charge model is established with four additional constant parameters reflecting the initial conditions of all dynamic elements. Thus, the line equilibrium point only related to memristor initial condition in the voltage–current domain is transformed into some determined equilibrium points, whose locations and stabilities are explicitly related to all four initial conditions. Consequently, the initial condition-dependent extreme multistability phenomenon, which has not been quantitatively analyzed in the voltage–current domain, can readily be investigated through evaluating these determined equilibrium points. Most important of all, the initial condition-dependent dynamical behaviors are formulated as the system parameter-dependent behaviors in the newly constructed flux–charge model and thus can be rigorously captured in a hardware equivalent realization circuit. Numerical simulations and experimental measurements reveal that the control of extreme multistability is successfully achieved in flux–charge domain, which is significant for seeking potential engineering applications of multistable memristive circuits.

Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria

Memristor-based nonlinear dynamical system easily presents the initial condition-dependent dynamical phenomenon of extreme multistability, i.e., coexisting infinitely many attractors, which has been received much attention in recent years. By introducing an ideal and active flux-controlled memristor into an existing hypogenetic chaotic jerk system, an interesting memristor-based chaotic system with hypogenetic jerk equation and circuit forms is proposed. The most striking feature is that this system has four line equilibria and exhibits the extreme multistability phenomenon of coexisting infinitely many attractors. Stability of these line equilibria are analyzed, and coexisting infinitely many attractors’ behaviors with the variations of the initial conditions are investigated by bifurcation diagrams, Lyapunov exponent spectra, attraction basins, and phased portraits, upon which the forming mechanism of extreme multistablity in the memristor-based hypogenetic jerk system is explored. Specially, unusual transition behavior of long term transient period with steady chaos, completely different from the phenomenon of transient chaos, can be also found for some initial conditions. Moreover, a hardware circuit is design and fabricated and its experimental results effectively verify the truth of extreme multistablity.

Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability

This paper presents a novel fifth-order two-memristor-based Chua’s hyperchaotic circuit, which is synthesized from an active band pass filter-based Chua’s circuit through replacing a nonlinear resistor and a linear resistor with two different memristors. This physical circuit has a plane equilibrium and therefore emerges complex phenomenon of extreme multistability. Based on the mathematical model, stability distributions of three nonzero eigenvalues in the equilibrium plane are exhibited, from which it is observed that four different stability regions with unstable saddle-focus, and stable and unstable node-focus are distributed, thereby leading to coexisting phenomenon of infinitely many attractors. Furthermore, extreme multistability depending on two-memristor initial conditions is investigated by bifurcation diagrams and Lyapunov exponent spectra and coexisting infinitely many attractors’ behavior is revealed by phase portraits and attraction basins. At last, a hardware circuit is fabricated and some experimental observations are captured to verify that extreme multistability indeed exists in the two-memristor-based Chua’s hyperchaotic circuit.

Hidden extreme multistability in memristive hyperchaotic system

By utilizing a memristor to substitute a coupling resistor in the realization circuit of a three-dimensional chaotic system having one saddle and two stable node-foci, a novel memristive hyperchaotic system with coexisting infinitely many hidden attractors is presented. The memristive system does not display any equilibrium, but can exhibit hyperchaotic, chaotic, and periodic dynamics as well as transient hyperchaos. In particular, the phenomenon of extreme multistability with hidden oscillation is revealed and the coexistence of infinitely many hidden attractors is observed. The results illustrate that the long term dynamical behavior closely depends on the memristor initial condition thus leading to the emergence of hidden extreme multistability in the memristive hyperchaotic system. Additionally, hardware experiments and PSIM circuit simulations are performed to verify numerical simulations.