Coexistence Of Attractors Research Articles

Persistence and coexistence of infinite attractors in a fractal Josephson junction resonator with unharmonic current phase relation considering feedback flux effect

Josephson junction resonators are the devices which exhibit complex behaviours as a consequence of their inductive properties. Even though the insulating medium between Josephson junctions (JJs) is normally considered homogeneous, the fact that lithography is used to form the layer, it has fractal substrates. Such JJs are identified as fractal Josephson junctions (FJJs). In this paper, a new chaotic oscillator based on memristor and FJJ has been investigated. Superconductor properties can dramatically change its operating points especially voltage and heat that are related to Josephson tunnelling. Some changes in the operating points can cause the Josephson tunnelling junctions to oscillate in different oscillation modes in very high frequencies. This can be achieved by considering the potential across the junction with its flux feedback. In order to model the magnetic flux effect, we use a memristor whose memductance function is considered as an exponential function. By varying the type of the bias current, we could observe the property of infinitely coexisting attractors in the memristor-fractal Josephson junction oscillator, which is considered as a rare phenomenon in physical circuits. The proposed Josephson-Memristor circuit model is developed, and its equilibrium points, bifurcation and Lyapunov exponents are computed. As an engineering application, modelling the trajectories of the moving object has been realized. First, the SURF algorithm, which is not affected by the scale and rotations of the object, is used in the images to identify an object that tracks the states of the proposed Josephson-Memristor circuit. In this way, the coordinates of the orbits on which the object moves were determined on the image. In order to reproduce the orbits of the specified object, the coordinate information of the object has been trained to the artificial neural network model and the orbits of the object have been reproduced.

Study on Bias Control of Memristor Multistablity System

In this paper, we study a memristor chaotic system with bias control. Based on the analysis of basic dynamic behavior, the hidden attractor coexistence and multi-stability of the non-equilibrium memristor chaotic system are verified, the bias control based on DC power control is studied, and the polarity control of the output variable is realized by using the change of parameters. The sinusoidal function is introduced as the bias periodic function, so that the system can realize signal polarity control only by changing the initial conditions. The analog circuit of memristor chaotic system is designed and verified by simulation..

Two-Memristor-Based Chaotic System With Infinite Coexisting Attractors

Chaotic systems with memristor are favored by academia because of diversity of dynamics. This brief reports a novel two-memristor-based 4D chaotic system. Numerical simulation shows that the system can yield infinite coexisting attractors. The generation mode of infinite coexisting attractors is investigated as well. Finally, the circuit and microcontroller-based implementation of two-memristor-based chaotic system are also given, corresponding experimental results fit well with numerical simulations.

Effects of Symmetric and Asymmetric Nonlinearity on the Dynamics of a Third-Order Autonomous Duffing–Holmes Oscillator

A generalized third-order autonomous Duffing–Holmes system is proposed and deeply investigated. The proposed system is obtained by adding a parametric quadratic termmx2to the cubic nonlinear term−x3of an existing third-order autonomous Duffing–Holmes system. This modification allows the system to feature smoothly adjustable nonlinearity, symmetry, and nontrivial equilibria. A particular attention is given to the effects of symmetric and asymmetric nonlinearity on the dynamics of the system. For the specific case ofm=0, the system is symmetric and interesting phenomena are observed, namely, coexistence of symmetric bifurcations, presence of parallel branches, and the coexistence of four (periodic-chaotic) and six (periodic) symmetric attractors. Form≠0, the system loses its symmetry. This favors the emergence of other behaviors, such as the coexistence of asymmetric bifurcations, involving the coexistence of several asymmetric attractors (periodic-periodic or periodic-chaotic). All these phenomena have been numerically highlighted using nonlinear dynamic tools (bifurcation diagrams, Lyapunov exponents, phase portraits, time series, frequency spectra, Poincaré section, cross sections of the attraction basins, etc.) and an analog computer of the system. In fact, PSpice simulations of the latter confirm numerical results. Moreover, amplitude control and synchronization strategies are also provided in order to promote the exploitation of the proposed system in engineering.

Finite Cascades of Pitchfork Bifurcations and Multistability in Generalized Lorenz-96 Models

In this paper, we study a family of dynamical systems with circulant symmetry, which are obtained from the Lorenz-96 model by modifying its nonlinear terms. For each member of this family, the dimension n can be arbitrarily chosen and a forcing parameter F acts as a bifurcation parameter. The primary focus in this paper is on the occurrence of finite cascades of pitchfork bifurcations, where the length of such a cascade depends on the divisibility properties of the dimension n. A particularly intriguing aspect of this phenomenon is that the parameter values F of the pitchfork bifurcations seem to satisfy the Feigenbaum scaling law. Further bifurcations can lead to the coexistence of periodic or chaotic attractors. We also describe scenarios in which the number of coexisting attractors can be reduced through collisions with an equilibrium.

Coexistence of continuous attractors with different dimensions for neural networks

This work briefly investigates the coexistence of continuous attractors in the neural networks with Rectified Linear Unit (RELU) transfer function. Memory is stored as a manifold of stable states, or a continuous attractor. Continuous attractors are some low-dimensional manifolds embedded in a high-dimensional state space. One neural network may possess more than one continuous attractors. More importantly, we found that these multiple continuous attractors may have different dimensional, some are 2-D plane attractors and others are 1-D line attractors. It is also an enlightenment to study the continuous attractors of high dimensional models.

In search of hyperchaos in a high dimensional unmagnetized quantum plasma

Abstract The hyperchaos and multistability of electron acoustic waves in a quantum plasma model comprising of nondegenerate cold and degenerate hot electrons and stationary ions are investigated. A six-dimensional dynamical system is constructed from the fluid equations of the model considering traveling wave transformation. The stability analysis of the system is done by finding out the equilibria in the inertia frame. It is interesting to investigate that though the novel system is conservative, it can produce hyperchaos for a set of associated parameters. We have also reported the coexistence of many hyperchaotic attractors as the system is extremely sensitive to the initials. The signature of hyperchaos and coexisting hyperchaos in a conservative quantum plasma system has never been reported before.

Synchronous Reluctance Motor with Load Vibration Perturbation: Analysis, Electronic Implementation and Adaptive Backstepping Sliding Mode Control

Dynamical analysis and chaos control of a synchronous reluctance motor (SynRM) with a constant and sinusoidal load torque are studied in this paper. The SynRM with load vibration perturbation exhibits pitchfork bifurcation, steady state behavior, double-scroll chaotic attractor and coexistence of attractors between steady state behavior and double-scroll chaotic attractors. Furthermore, an analog circuit is designed and implemented in OrCAD-PSpice software to validate the dynamical behaviors found in the SynRM with load vibration perturbation during the numerical simulations. Finally, an adaptive backstepping sliding mode control is used to suppress the chaotic oscillations in the SynRM with load vibration perturbation. Numerical simulations confirm the effectiveness of the proposed controller.

Dynamics of a Euler–Bernoulli beam on nonlinear viscoelastic foundations: a parameter space analysis

The dynamics and spectrum of vibrations of an uniform Euler–Bernoulli beam on nonlinear viscoelastic foundations are investigated in parameter space. An alternative method of approach based on mean-field approximation is proposed. Spatial and temporal solution are obtained in an interactive way. The spectrum is reached directly from spatial solutions. It was shown that the system exhibits sudden birth and death of chaotic attractors (crises) and that there are periodic windows immersed in chaotic regions. It was determined the regions in the parameter space where chaotic and regular attractors occur. Moreover, the model presents the coexistence of many attractors. The numerical simulations found geometrical structures known as shrimps, and these periodic structures are involved by chaotic regions. In this work it was investigated the dynamics of a beam of Euler–Bernoulli with viscous and dynamic support. It is presented, then, a model to treat this type of problem, which is nonlinear and non-autonomous. The dynamics of the model presents a complex behavior, being in some cases sensitive to the initial conditions. Changing properly the parameters the dynamics goes from chaotic behavior to periodic regime. As the system is forced and dissipative, complex structures in the parameter space, known as shrimps, arise. Also, it was found that the spatial solution is not sensitive to the dynamics, although they are coupled. However, the dynamics is very sensitive to the mean quadratic deformation of the beam and, therefore, it is related to the vibration modes obtained in the spatial solution.

Dynamics and Complexity Analysis of Fractional-Order Chaotic Systems with Line Equilibrium Based on Adomian Decomposition

In this paper, the Adomian decomposition method (ADM) is applied to solve the fractional-order system with line equilibrium. The dynamics of the system is analyzed by means of the Lyapunov exponent spectrum, bifurcations, chaotic attractor, and largest Lyapunov exponent diagram. At the same time, through the Lyapunov exponent spectrum and bifurcation graph of the system under the change of the initial value, the influence of fractional order q on the system state can be observed. That is, integer-order systems do not have the phenomenon of attractors coexistence, while fractional-order systems have it.

Asymmetry-Induced Dynamics for a Class of Diode-Based Chaotic Circuits: A Case Study

We consider the modeling and asymmetry-induced dynamics for a class of chaotic circuits sharing the same feature of an antiparallel diodes pair as the nonlinear component. The simple autonomous jerk circuit of [J. Kengne, Z. T. Njitacke, A. N. Nguomkam, M. T. Fouodji and H. B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit, Int. J. Bifurcation Chaos Appl. Sci. Eng. 26 (2016) 1650081] is used as the prototype. In contrast to current approaches where the diodes are assumed to be identical (and thus a perfect symmetric circuit), we examine the more realistic situation where the diodes have different electrical properties in spite of unavoidable scattering of parameters. In this case, the nonlinear component formed by the diodes pair displays an asymmetric current–voltage characteristic which induces asymmetry of the whole circuit. The model is described by a continuous-time 3D autonomous system (ODEs) with exponential nonlinearities. We examine the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectory plots as the main indicators. Period doubling route to chaos, merging crisis, and multiple coexisting (i.e., two, four, or six) mutually symmetric attractors are reported in the symmetric mode of oscillation. In the asymmetric mode, several unusual nonlinear behaviors arise such as coexisting bifurcations, hysteresis, asymmetric double-band chaotic attractor, crisis, and coexisting multiple (i.e., two, three, four, or five) asymmetric attractors for some suitable ranges of parameters. Theoretical analyses and circuit experiments show a very good agreement. The results obtained in this work let us conjecture that chaotic circuits with antiparallel diodes pair are capable of much more complex dynamics than what is reported in the current literature and thus should be reconsidered accordingly in spite of the approach followed in this work.

Asymmetric memristive Chua’s chaotic circuits

ABSTRACT Asymmetric property of the pinched hysteresis loop usually appears in a real memristor. To imitate this property, an asymmetric memristor emulator is constructed by an analog circuit consisting of a non-balance diode-bridge and a resistor-capacitor (RC) filter. By coupling this emulator to the classical Chua’s circuit in parallel in both forward and reverse ways, two asymmetric memristive Chua’s circuits are proposed and their mathematical models are established. Because of the symmetry between the forward and reverse circuits, the forward memristive Chua’s (FAMC) circuit is focused on modelling the normalised system and investigating the equilibrium stability. With the FAMC system, several numerical plots are employed to discuss the parameter-dependent complex dynamics and initial-dependent coexisting behaviours. The numerical results manifest that there are three equilibrium points in the FAMC system, among which two non-zero equilibrium points are asymmetric, resulting in the emergence of asymmetric attractors. With the change of system parameters, the stability of one non-zero equilibrium point converts from unstable saddle-focus into stable node-focus, leading to the coexistence of multiple attractors. In addition, hardware experiments on a breadboard level are executed and the captured results well verify the numerical plots.

Nonlinear Dynamics in a Chemical Reaction under an Amplitude-Modulated Excitation: Hysteresis, Vibrational Resonance, Multistability, and Chaos

This paper deals with the effects of an amplitude-modulated (AM) excitation on the nonlinear dynamics of reactions between four molecules. The computation of the fixed points of the autonomous nonlinear chemical system has been made in detail using the Cardan’s method. Hopf bifurcation has been also successfully checked. Routes to chaos have been investigated through bifurcations structures, Lyapunov exponent, phase portraits, and Poincaré section. The effects of the control force on chaotic motions have been strongly analyzed, and the control efficiency is found in the cases g=0 (unmodulated case) and g≠0 with Ω=ω and Ω/w≠p/q; p and q are simple positive integers. Vibrational resonance (VR), hysteresis, and coexistence of several attractors have been studied in detail based on the relationship between the frequencies of the AM force. Results of analytical investigations are validated and complemented by numerical simulations.

Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model

AbstractOver 50 years since Lorenz’s 1963 study and a follow-up presentation in 1972, the statement “weather is chaotic” has been well accepted. Such a view turns our attention from regularity associated with Laplace’s view of determinism to irregularity associated with chaos. In contrast to single-type chaotic solutions, recent studies using a generalized Lorenz model (GLM) have focused on the coexistence of chaotic and regular solutions that appear within the same model using the same modeling configurations but different initial conditions. The results, with attractor coexistence, suggest that the entirety of weather possesses a dual nature of chaos and order with distinct predictability. In this study, based on the GLM, we illustrate the following two mechanisms that may enable or modulate two kinds of attractor coexistence and, thus, contribute to distinct predictability: 1) the aggregated negative feedback of small-scale convective processes that can produce stable nontrivial equilibrium points and, thus, enable the appearance of stable steady-state solutions and their coexistence with chaotic or nonlinear oscillatory solutions, referred to as the first and second kinds of attractor coexistence; and 2) the modulation of large-scale time-varying forcing (heating) that can determine (or modulate) the alternative appearance of two kinds of attractor coexistence. Based on our results, we then discuss new opportunities and challenges in predictability research with the aim of improving predictions at extended-range time scales, as well as subseasonal to seasonal time scales.

Unfolding Nonlinear Dynamics in Analogue Systems With Mem-Elements

The paper considers a relevant class of networks containing memristors and (possibly) nonlinear capacitors and inductors. The goal is to unfold the nonlinear dynamics of these networks by highlighting some main features that are potentially useful for real-time signal processing and in-memory computing. In particular, an analytic treatment is provided for dynamic phenomena as the presence of invariant manifolds, the coexistence of different regimes, complex dynamics and attractors and the phenomenon of bifurcations without parameters, i.e., bifurcations due to changing the initial conditions of the state variables for a fixed set of circuit parameters. The paper also addresses the issue of how to design pulse independent voltage or current sources to steer the network dynamics through different manifolds and attractors. Two relevant examples are worked out in details, namely, a variant of Chua's circuit with a memristor and a nonlinear capacitor and a relaxation oscillator with a memristor and a nonlinear inductor. In the latter example, the paper also studies the effect on manifolds and coexisting dynamics when real memristive devices are accounted for using a class of extended memristor models. The analysis is conducted by means of a recently developed technique named flux-charge analysis method (FCAM). Numerical simulations are presented to confirm the theoretic findings.

Complex dynamics of some models of antimicrobial resistance on complex networks

Studying the spread of epidemics and diseases is a worldwide problem especially during the current time where the whole world is suffering from COVID‐19 pandemic. Antimicrobial resistance (AMR) and waning vaccination are classified as worldwide problems. Both depend on the exposure time to antibiotic and vaccination. Here, a simple model for competition between drug‐resistant and drug‐sensitive bacteria is given. Conditions for local stability are investigated, which agree with observation. Existence of positive solution in the AMR complex networks is proved. Dynamics of the identical AMR models are explored with different topologies of complex networks such as global, star, line, and unidirectional line networks coupled through their susceptible states. Chaotic attractors are shown to exist as the AMR models are located on all these topologies of complex networks. Thus, it is found that the dynamics of the AMR model become more complicated as it is located on either integer‐order or fractional‐order complex networks. Furthermore, a discretized version of the fractional AMR model is presented. Complex dynamics such as existence of Neimark–Sacker, flip bifurcations, coexistence of multiple attractors, homoclinic connections, and multiple closed invariant curves are investigated. Basin sets of attraction are also computed. Finally, the discretized system is located on complex networks with different topologies which show rich variety of complex dynamics. Also, 0–1 test is used to verify the existence of unpredictable dynamics. So, studying the dynamics of AMR models on complex networks is very helpful to understand the mechanism of spread of diseases.

A Calculation Method on Bifurcation and State Parameter Sensitivity Analysis of Piecewise Mechanical Systems

The paper proposes a novel calculation method on the sensitivity analysis of bifurcation parameters and states in a single-degree-of-freedom (SDoF) impacting system. It presents the causes to (non-) smooth bifurcations in virtue of parameter sensitivity analysis. The derivation of the system’s Poincaré mappings is used to integrate the Floquet matrix. It performs the identifications of the main and the most sensitive bifurcation parameters by disturbing the eigenvalues of Floquet matrix. Moreover, the ones that have appreciable effect on the dynamic characteristics of the system can be effectively identified from entire bifurcation parameters and states. The coexistence of distinct attractors is demonstrated with the parameter sensitivity analysis on the Floquet matrix associated with bifurcation parameters and states. It also considers the cooperating dynamic performance on the two-dimensional parameter space by varying different pairs of bifurcation parameters.

Network Dynamics of a Fractional-Order Phase-Locked Loop with Infinite Coexisting Attractors

We have investigated a fractional-order phase-locked loop characterised by a third-order differential equation. The integer-order mathematical model of the phase-locked loop (PLL) is first converted to fractional order using the Caputo-Fabrizio method. The stability of the equilibrium points is discussed in detail in both parameter and fractional-order domain. The proposed fractional-order phase-locked loop (FOPLL) model shows multiple coexisting attractors which was not discussed in the earlier literature of PLL. The significance of these infinite coexisting attractors is that they exist in the operation region of the PLL between [−π,π] which increases the complexity of operation of the PLLs. Mainly when such FOPLLs are used in large-scale networks, the synchronisation of the FOPLLs becomes complicated and will result in unstable control conditions. Hence, studying the network dynamics of such FOPLLs is significant which motivates us to investigate the synchronisation phenomenon of the FOPLLs constructed in a square network. We could show that, because of the multiple coexisting attractors, the FOPLLs show various synchronisation phenomena, and more importantly in the chaotic region for lower fractional-order values, we could show that the FOPLLs are synchronised and this finding is very useful to completely analyse the FOPLL networks in high-frequency operations.

I love shopping…but what am I going to buy? Social interaction and consumption choices.

In the present work, we analyze the emergence of fashion cycles and complex phenomena in a discrete time dynamic model in which a population is divided into two groups, bandwagoners and snobs. Both groups react differently to the aggregate demand for goods in the previous period and, in particular, bandwagoners imitate the consumption styles prevailing in society, while snobs try to distinguish themselves from them. We consider a first specification of the model in which the structure of the population is fixed and we show how in the case of polymorphic population it is possible to observe cyclical behaviors in collective consumption and the onset of chaotic regimes. We further propose an extension of the model in which we investigate the interdependence between the evolution of collective consumption choices and the evolution of the structure of the groups in a framework in which individuals may change the structure of preferences and then switch between being of one type or another. In the extension, we analyze how both consumption cycles (and then fashion cycles) and the evolution of the groups may lead to the emergence of chaotic dynamics, as well as the coexistence of attractors.

Design and implementation of a new memristive chaotic system with application in touchless fingerprint encryption

In this paper, a new memristive chaotic system was constructed from three-dimensional Lorenz-type system. The system has infinitely many equilibria and exhibits coexisting attractors. The dynamic evolution corresponds to the parameters and the coupling strength indicate that the system is easy to generate chaos. Also the bifurcation diagrams from different initial conditions determine the coexistence of multiple attractors. An electronic circuit is done for verifying the physically existence of the system. Based on this system, chaos-based random number generator and the corresponding randomness tests are studied. An algorithm for touchless fingerprint encryption is established. Some comparative tests illustrate the effectiveness of the algorithm.