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This paper proposes a new control method applicable for a class of non-autonomous dynamical systems that naturally exhibit coexisting attractors. The central idea is based on knowledge of a system's basins of attraction, with control actions being applied intermittently in the time domain when the actual trajectory satisfies a proximity constraint with regards to the desired trajectory. This intermittent control uses an impulsive force to perturb one of the system attractors in order to switch the system response onto another attractor. This is carried out by bringing the perturbed state into the desired basin of attraction. The method has been applied to control both smooth and non-smooth systems, with the Duffing and impact oscillators used as examples. The strength of the intermittent control force is also considered, and a constrained intermittent control law is introduced to investigate the effect of limited control force on the efficiency of the controller. It is shown that increasing the duration of the control action and/or the number of control actuations allows one to successfully switch between the stable attractors using a lower control force. Numerical and experimental results are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear. [ABSTRACT FROM AUTHOR]

Abstract: A network of neurons with dendritic dynamics is analyzed in this paper. Two stable regimes of the complete network can coexist under continuous weak stimulation: the oscillatory synchronized regime and the quiet regime, where all neurons stop firing completely. It is shown that a single control pulse can calm a single neuron as well as the whole network, and the network stays in the quiet regime as long as the weak stimulation is turned on. It is also demonstrated that the same control technique can be effectively used to calm a random Erdös–Renyi network of dendritic neurons. Moreover, it appears that the random network of dendritic neurons can evolve into the quiet regime without applying any external pulse-based control techniques. [Copyright &y& Elsevier]

This paper is devoted to the analysis of a new simple rational map of the plane. Its dynamics are described in some detail, along with some other dynamical phenomena. In particular, the map under consideration is the first simple rational map whose fraction has no vanishing denominator that gives chaotic attractors via a quasi-periodic route to chaos. [ABSTRACT FROM AUTHOR]

In this work the strange behavior of an impact oscillator with a one-sided elastic constraint discovered experimentally is compared with the predictions obtained using its mathematical model. Extensive experimental investigations undertaken on the rig developed at the Aberdeen University reveal different bifurcation scenarios under varying excitation frequency near grazing which were recorded for a number of values of the excitation amplitude. In the paper, particular attention is paid to the chaotic oscillations recorded near grazing frequency when a nonimpacting orbit becomes an impacting one under increasing excitation frequency. It was found that the evolution of the attractor is governed by a complex interplay between smooth and nonsmooth bifurcations, and the interactions between a number of coexisting orbits. The occurrence of coexisting attractors is manifested in the experimental results through discontinuous transitions from one orbit to another via boundary crisis. In some cases, the basins of attraction have a fractal structure. Detailed numerical exploration also revealed coexisting unstable periodic orbits. These stable and unstable coexisting orbits are often born far from the parameter values at which they influence the system dynamics. The very rich dynamics of the bilinear oscillator close to grazing is demonstrated and typical mechanisms of the attractors' appearance and disappearance are explained using stability analysis. [ABSTRACT FROM AUTHOR]

Abstract: The coexistence of multiple attractors in the Koper–Gaspard model of the electrochemical oscillatory reactions on the rotating-disk electrode (RDE) has already been investigated. It is held that the Koper–Gaspard model dynamics has many multiple attractors between simple harmonic oscillations or chaos and mixed-mode oscillations in many ranges of two-dimensional parametric space. Qualitative comparison of multiple attractors demonstrates bizarre winding of trajectories round themselves due to no-intersection theorem. Two- and three-dimensional basins of attraction have been scrutinized showing complex stripped-pattern basins which make predictability difficult in some regions of basins. Physico-chemical interpretation of the results has been discussed according to the feedback loops of electrochemical variables. Physical interpretation of the model multiple attractors implies that electrochemical oscillatory reactions are very sensitive dependence on initial values of electrochemical variables such as applied voltage or concentrations of electroactive species. This may change the entire pattern of behavior by slight shifting of initial conditions into the other basin. Such complex phenomena may provide sensible unpredictability for electrochemical time series. [Copyright &y& Elsevier]

Abstract: In this paper, the dynamical model of the shaker system, Poincaré maps, Jacobian matrix and power spectrum are established. Different phase-locking phenomena and chaotic bubbles are investigated in the vicinity of three coexisting critical points including Hopf–Hopf bifurcation point, 1:3 resonance point and 1:4 resonance point. In two strong resonance cases, phase-locking dynamics and associated bifurcations are easily to occur. Coexisting attractors have also been introduced to provide mechanisms for chaotic bubbles with connections between pieces. The occurrence of phase locking on doubling torus to multi-period leads to interruption of torus-doubling bifurcation. Isolated chaotic bubbles are birth via period-doubling bifurcation of such a multi-period. Phase-locking phenomena on T2 torus are also observed in such a neighborhood of critical points. The number of periods on torus by phase locking can be identified by power spectrum methods. The system parameters may be optimized by studying of phase-locking dynamics of this system. [Copyright &y& Elsevier]

We study a delayed system with feedback modulation of the nonlinear parameter. Study of the system as a function of nonlinearity and modulation parameters reveals complex dynamical phenomena: different types of coexisting attractors, local or global bifurcations and transitions. Bistability and dynamical attractors can be distinguished in some parameter-space regions, which may be useful to drive chaotic dynamics, unstable attractors or bistability towards regular dynamics. At the bifurcation to bistability, two striking features are that they lead to fundamentally unpredictable behavior of orbits and crisis of attractors as system parameters are varied slowly through the critical curve. Unstable attractors are also investigated in bistable regions, which are easily mistaken for true multi-periodic orbits judging merely from zero measure local basins. Lyapunov exponents and basins of attraction are also used to characterize the phenomenon observed. [ABSTRACT FROM AUTHOR]

This paper studies a control method for switching stable coexisting attractors of a class of non-autonomous dynamical systems. The central idea is to introduce a continuous path for the system's trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information of the desired attractor. The behaviour of the control is studied numerically for both non-smooth and smooth dynamical systems, using a soft-impact and a Duffing oscillators as examples. Special attention is given to identify the parametric regions where the proposed control strategy is applicable by using path-following methods implemented via the continuation platform COCO. It is shown that the proposed control concept can be implemented through either using an external control input or varying a system parameter. Finally, extensive numerical results are presented to validate the proposed control methods. • Control of coexisting attractors in non-autonomous dynamical systems is studied. • It is achieved by either using an external control or varying a system parameter. • It is applied numerically to both a soft-impact system and a Duffing oscillator. • Extensive numerical results are presented to validate its control effectiveness. • Path-following methods are adopted to identify the effective control regions. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a new numerical method to compute the separatrix of the basins of attraction of coexisting attractors in a forced friction oscillator. Numerical results show that its intersection with a Poincaré section is a non-smooth curve. [Copyright &y& Elsevier]

In a simple model of financial market dynamics, we allow the price of a risky security to be set by a market maker depending on the excess demand of heterogeneous interacting traders, fundamentalists and chartists, who place their orders based upon different expectations schemes about future prices: while chartists rely on standard trend-based rules, fundamentalists are assumed to know the economic environment and to form their beliefs accordingly. As price moves away from the long-run fundamental, fundamentalists become less confident in their forecasts, and put increasing weight on a reversion towards the fundamental price. The resulting two-dimensional discrete time dynamical system can exhibit a rich range of dynamic scenarios, often characterized by coexistence of attractors. A simple noisy version of the model reveals a variety of possible patterns for return time series. [ABSTRACT FROM AUTHOR]

Abstract: A simple asset pricing model with two types of boundedly rational traders, fundamentalists and chartists, is studied. Fractions of trader types change over time according to evolutionary learning, with chartists conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously and two generic mechanisms are proposed as an explanation: (1) coexistence of a stable steady state and a stable limit cycle, due to a so-called Chenciner bifurcation of the system and (2) intermittency and associated bifurcation routes to strange attractors. Economic intuition as to why these phenomena arise in nonlinear multi-agent evolutionary systems is provided. [Copyright &y& Elsevier]

The coexistence of several stable states for a given set of parameters has been observed in many natural and experimental systems as well as in theoretical models. This paper gives an overview over the wide range of applications in different disciplines of science. Furthermore, different system classes possessing multistability are analyzed in terms of the appearance of coexisting attractors and their basins of attraction. It is shown that multistable systems are very sensitive to perturbations leading to a noise-induced hopping process between attractors. The role of chaotic saddles in the escape from attractors in multistable systems is discussed. [ABSTRACT FROM AUTHOR]

We report on a numerical investigation of the nonlinear dynamics of a passively mode-locked fiber laser containing a long period fiber grating. The model is based on the normalized complex Ginzburg-Landau equation and the nonlinear coupled mode equations of the grating. The numerical results indicate the existence of passive mode-locking and autosoliton generation in the cavity of the laser. Both single and bound soliton pulse trains exhibit period doubling bifurcations and a route to chaos as the normalized saturated gain is increased. Furthermore, we show the presence of long period pulsation, soliton sidebands and possible coexisting attractors excited by multisoliton formation and soliton energy quantization. [ABSTRACT FROM AUTHOR]

It is shown that even simple models of angular motion transfer mechanisms possess features characteristic to non-linear dynamical systems: coexistence of periodic attractors, non-synchronous motion modes, chaotic solutions. Some of those features are quite unique. The transition to chaos takes place when the damping of the output element is increased. The basin boundaries of strange attractors are not fractal. The existence of multiple stable attractors enables the design of a new type of control strategy for angular motion transfer mechanisms. [ABSTRACT FROM AUTHOR]

This paper reports a simple parallel chaotic circuit with only four circuit elements: a capacitor, an inductor, a thermistor, and a linear negative resistor. The proposed system was analyzed with MATLAB R2018 through some numerical methods, such as largest Lyapunov exponent spectrum (LLE), phase diagram, Poincaré map, dynamic map, and time-domain waveform. The results revealed 11 kinds of chaotic attractors, 4 kinds of periodic attractors, and some attractive characteristics (such as coexistence attractors and transient transition behaviors). In addition, spectral entropy and sample entropy are adopted to analyze the phenomenon of coexisting attractors. The theoretical analysis and numerical simulation demonstrate that the system has rich dynamic characteristics. [ABSTRACT FROM AUTHOR]

The mathematical modeling of memristor in discrete-time domain is an attractive new issue, but there are still some problems to be explored. This paper studies an interesting second-order memristor-based map model, and the model is constructed to three systems based on Caputo fractional-order difference. Their dynamic behaviors are investigated by the volt–ampere curve, bifurcation diagram, maximum Lyapunov exponent, attractor phase diagram, complexity analysis and basin of attraction. Numerical simulation analysis shows that the fractional-order system exhibits quasi periodic, chaos, coexisting attractors and other complex behaviors, which demonstrates more abundant dynamic behaviors of the fractional-order form. It lays a good foundation for the future analysis or engineering application of the discrete memristor. • The fractional-order difference is introduced to the discrete memristor map. • Three types of fractional-order memristor systems are designed according to the position of the fractional-order difference. • Complex dynamics including coexisting attractor are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order memristive model with infinite coexisting attractors is investigated. The numerical solution of the system is derived based on the Adomian decomposition method (ADM), and its dynamic behaviors are analyzed by means of phase diagrams, bifurcation diagrams, Lyapunov exponent spectrum (LEs), dynamic map based on SE complexity and maximum Lyapunov exponent (MLE). Simulation results show that it has rich dynamic characteristics, including asymmetric coexisting attractors with different structures and offset boosting. Finally, the digital signal processor (DSP) implementation verifies the correctness of the solution algorithm and the physical feasibility of the system. [ABSTRACT FROM AUTHOR]

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results. [ABSTRACT FROM AUTHOR]

In this paper, a chaotic three dimansional dynamical system is proposed, that is a modification of the system in Volos et al. (2017). The new system has two hyperbolic sine nonlinear terms, as opposed to the original system that only included one, in order to optimize system's chaotic behavior, which is confirmed by the calculation of the maximal Lyapunov exponents and Kaplan-Yorke dimension. The system is experimentally realized, using Bi-color LEDs to emulate the hyperbolic sine functions. An extended dynamical analysis is then performed, by computing numerically the system's bifurcation and continuation diagrams, Lyapunov exponents and phase portraits, and comparing the numerical simulations with the circuit simulations. A series of interesting phenomena are unmasked, like period doubling route to chaos, coexisting attractors and antimonotonicity, which are all verified from the circuit realization of the system. Hence, the circuit setup accurately emulates the chaotic dynamics of the proposed system. [ABSTRACT FROM AUTHOR]

This article proposes a new fractional-order discrete-time chaotic system, without equilibria, included two quadratic nonlinearities terms. The dynamics of this system were experimentally investigated via bifurcation diagrams and largest Lyapunov exponent. Besides, some chaotic tests such as the 0–1 test and approximate entropy (ApEn) were included to detect the performance of our numerical results. Furthermore, a valid control method of stabilization is introduced to regulate the proposed system in such a way as to force all its states to adaptively tend toward the equilibrium point at zero. All theoretical findings in this work have been verified numerically using MATLAB software package. [ABSTRACT FROM AUTHOR]

Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called "self-excited attractors" has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called "hidden attractors", which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system. [ABSTRACT FROM AUTHOR]

• A new 3-D asymmetric chaotic system with only one quadratic nonlinearity is proposed. • This system shows the bi-stability phenomenon of coexisting asymmetric attractors. • The amplitude, frequency, and polarity of the generated signals are freely controlled. • The pseudo-multi-scroll attractors are generated from the chaotic system via non-autonomous approach. The design and circuit implementation of flexible chaotic systems with desired properties have been of great interest due to their promising applications in various chaos-based information systems. In this paper, a simple three-dimensional autonomous chaotic system with only one quadratic nonlinearity and six linear terms is proposed. The system is asymmetric, which has two equilibria and can generate a single-scroll attractor. This system exhibits asymmetric bi-stability behaviors of chaotic attractors coexisting with periodic limit cycles and other chaotic attractors. By introducing control functions to coefficients of specific linear term or quadratic term, the system can be extended as an amplitude-controllable, equilibrium-shiftable chaotic system. Consequently, attractor rotation is implemented via polarity symmetry, and pseudo-multi-scroll attractors are further generated via nonautonomous approach. The theoretical analyses are confirmed by both the numerical simulations and circuit verifications. [ABSTRACT FROM AUTHOR]

Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems. [ABSTRACT FROM AUTHOR]

On the basis of the bistable bi-local active memristor (BBAM), an active memristor (AM) and its emulator were designed, and the characteristic fingerprints of the memristor were found under the applied periodic voltage. A memristor-based chaotic circuit was constructed, whose corresponding dynamics system was described by the 4-D autonomous differential equations. Complex dynamics behaviors, including chaos, transient chaos, heterogeneous coexisting attractors, and state-switches of the system were analyzed and explored by using Lyapunov exponents, bifurcation diagrams, phase diagrams, and Poincaré mapping, among others. In particular, a novel exotic chaotic attractor of the system was observed, as well as the singular state-switching between point attractors and chaotic attractors. The results of the theoretical analysis were verified by both circuit experiments and digital signal processing (DSP) technology. [ABSTRACT FROM AUTHOR]

In this paper, a new voltage-controlled memristor is presented. The mathematical expression of this memristor has an absolute value term, so it is called an absolute voltage-controlled memristor. The proposed memristor is locally active, which is proved by its DC V–I (Voltage–Current) plot. A simple three-order Wien-bridge chaotic circuit without inductor is constructed on the basis of the presented memristor. The dynamical behaviors of the simple chaotic system are analyzed in this paper. The main properties of this system are coexisting attractors and multistability. Furthermore, an analog circuit of this chaotic system is realized by the Multisim software. The multistability of the proposed system can enlarge the key space in encryption, which makes the encryption effect better. Therefore, the proposed chaotic system can be used as a pseudo-random sequence generator to provide key sequences for digital encryption systems. Thus, the chaotic system is discretized and implemented by Digital Signal Processing (DSP) technology. The National Institute of Standards and Technology (NIST) test and Approximate Entropy analysis of the proposed chaotic system are conducted in this paper. [ABSTRACT FROM AUTHOR]