Coexistence Of Multiple Attractors Research Articles

Collective dynamics of two coupled Hopfield inertial neurons with different activation functions: theoretical study and microcontroller implementation

This work deals with the regular and chaotic dynamics of a system made up of two Hopfield-type neurons with two different activation functions: the hyperbolic tangent function and the Crespi function. The mathematical model is in the form of an autonomous differential system of order four with odd symmetry. The analysis highlights nine equilibrium points and four of these points experience a Hopf bifurcation at the same critical value of a control parameter which can be either the diss1ipation parameter or one of the coupling coefficients. This makes plausible the presence of four parallel bifurcation branches as well as the coexistence of multiple attractors in the behavior of the system. One of the highlights revealed in this work is the coexistence of three double-scroll type attractors of particular topology as well as the presence of a four-spiral attractor. Furthermore, the coexistence of both self-excited and hidden dynamics is also reported. All this plethora of dynamics is elucidated by making use of the usual tools for analyzing nonlinear systems such as bifurcation diagrams, the maximum of Lyapunov exponent, basins of attractions as well as phase portraits. A physical implementation of the microcontroller-based system is envisaged in order to confirm the plethora of behaviors observed theoretically.

Dynamic Analysis and DSP Implementation of Memristor Chaotic Systems with Multiple Forms of Hidden Attractors

In this paper, a new six dimensional memristor chaotic system is designed by combining the chaotic system with a memristor. By analyzing the phase diagram of the chaotic attractors, eleven different attractors are found, including a multi-wing attractor and symmetric attractors. By analyzing the equilibrium point of the system, it is proven that the system has the property of a hidden chaotic attractor. The dynamic behavior of the system when the three parameters change is analyzed by means of LEs and a Bifurcation diagram. Other phenomenon, such as chaos degradation, coexistence of multiple attractors and bias boosting, are also found. Finally, the simulation on the DSP platform also verifies the accuracy of the chaotic system simulation. The theoretical analysis and simulation results show that the system has rich dynamical characteristics; therefore, it is suitable for secure communication and image encryption and other fields.

A Novel four - Wing chaotic system with multiple attractors based on hyperbolic sine: Application to image encryption*

A novel symmetrical four-wing fourth-order chaotic system is constructed. The basic dynamic characteristics of the system were analyzed by phase diagram, bifurcation diagram, Lyapunov index spectrum, Poincare cross section diagram and 0–1 test. In addition, the chaotic attractors under different parameters in the system are analyzed. In the dynamic analysis of the new system, it is found that the new system has some characteristics, such as multi-stability, offset boosting, multi-state transition phenomenon, transient chaos and intermittent chaos, and coexistence of multiple attractors. These features have the value of in-depth analysis compared to previous systems and can make it promising for more applications. Moreover, after calculating the complexity of the system by C0 algorithm at different initial values, it is found that the complexity of the system has been stable. Due to the existence of many characteristics, the new chaotic system has attracted great attention. At the same time, the circuit design of the system is realized by using Multisim simulation software and FPGA digital hardware circuit. Finally, a novel and efficient image encryption algorithm is designed by combining multi-direction pixel scrambling and DNA dynamic encryption. The NIST test, key space, encryption histogram, adjacent pixel correlation, robustness and information entropy are analyzed by encrypting images using chaotic sequences of the new system. In a word, its plentiful characteristics and intricate phenomena have significant reference value in the field of chaotic image encryption.

A Simple memristive chaotic system with complex dynamics and ITS image encryption application

In this paper, a new memristive chaotic system with a relatively simple structure and complex dynamic behavior is proposed by introducing an absolute memristor into the VB19 chaotic system as a positive feedback term. The analysis shows that the new system is a fractional dimensional dissipative system and is unstable at the line equilibrium point. The system’s dynamic characteristics are studied using a bifurcation diagram and Lyapunov exponent’s spectrum. The numerical simulation shows that the period-doubling bifurcation phenomenon occurs when changing the parameters and system change from period to chaos. When the initial condition changes, the dynamic evolution of the system indicates that the coexistence of multiple attractors and chaotic signals can be controlled by local amplitude. In addition, a new image encryption algorithm based on the system is designed and the performance of this algorithm is tested through numerical analyses. The system’s analog circuit is built using electronic components and also realized by the microprocessor. The circuit realization results consistent with the numerical simulation results are obtained to verify the physical existence of the system.

Coexistence of Multiple Attractors in a Novel Simple Jerk Chaotic Circuit With CFOAs Implementation

A novel, simple Jerk chaotic circuit with three current feedback operational amplifiers included (CFOA-JCC) is proposed, which has a simpler circuit structure, fewer components, but higher frequency characteristics. The dynamic behaviors of CFOA-JCC are analyzed, including equilibrium, stability, Lyapunov exponent, bifurcation diagram, offset boosting, and phase diagram. Furthermore, the frequency spectrum characteristic of the ordinary op-amps Jerk chaotic circuit was compared with CFOA-JCC under the same circuit parameters, and the chaotic attractor frequency of CFOA-JCC can reach about 650 kHz, much better than that of ordinary op-amps (12 kHz). Numerical simulation shows that CFOA-JCC has coexisting attractors, verified by hardware circuit experiments.

Attenuation of Line Spectra and Vibration Based on a Chaotification Method and Global Analysis

The chaotification method can be used to weaken the line spectra component of the ship's radiated noise. The main obstacle that plagues the engineering realization of this innovative concept is how to realize the small-amplitude chaotic motion of the nonlinear vibration isolation system (NVIS) in a large parameter range and how to maintain the chaotic state under variable operating conditions. In this paper, the dimensionless dynamic equation of the NVIS with a flexible base is derived. A control method similar to generalized synchronization is used to realize the continuous chaos of the system. The maximum Lyapunov exponent, the conditional Lyapunov exponent, and disturbances are introduced to verify the effectiveness of the control method and its stability and robustness in a large range of parameters. Aiming at the coexistence of multiple attractors, a new global analysis method is proposed, which is used to study the distribution of basin of attraction (BA). According to the changes of BA before and after the system is controlled, the BA is partitioned. Initial conditions from different partitions are selected to study the attenuation effect of line spectra and vibration, and the favorable initial conditions for realizing the small-amplitude chaotic motion of the system are given. The method may be extended to the global analysis of other controlled systems.

Rare and hidden attractors in a periodically forced Duffing system with absolute nonlinearity

• The composite cell coordinate system (CCCS) method is modified to non-smooth systems. • Rare and hidden attractors of a non-smooth Duffing system are studied. • The accurate global properties such as attractor and basin can be obtained. • Hidden and rare attractors can be controlled by adjusting the amplitude. • The results show the effectiveness of our proposed improvement strategy. Hidden attractors and rare attractors are two kinds of special attractors in multistable systems, studying the appearance and properties of hidden attractors and rare attractors can increase the possibility of the system remaining on the ideal attractor and reduce the risk of sudden jump to unexpected behavior. This paper presents an investigation of the rare attractors and hidden attractors of a nonautonomous Duffing-like system with absolute function. Based on modified composite cell coordinate system (CCCS) method, the attractors and the corresponding basins are shown and the coexistence of multiple attractors is observed. By calculating the probability that the system is on each attractor, we discover different rare attractors in this system whose basins of attraction are extremely small. In addition, the equilibria of the system are calculated for a given set of parameters, according to the property that the hidden attractor does not contain any equilibrium, several hidden attractors are found for three parameter ranges. With the change of the amplitude of external excitation, the rare and hidden attractors appear in a very small parameter range. The results show that the modified CCCS method is a powerful tool to research the hidden and rare attractors in the non-smooth system.

A novel chaotic circuit with coexistence of multiple attractors and state transition based on two memristors

A novel chaotic circuit that includes two memristors, an inductor, and a capacitor in parallel is planned, then the dimensionless mathematical model is built. Seven kinds of different attractors are found in the system. Using traditional dynamical analysis methods, the circuit system's equilibrium point and stability are analyzed, and under certain conditions, this system not only has infinite equilibrium points but also has no equilibrium point. Then, the dynamical behaviors with the changing parameter of the system are analyzed in-depth. A wealth of special phenomena has been discovered, such as the multi-state transition and the coexisting attractors. At last, the circuit system is implemented by using the DSP platform, and the theoretical analysis is verified by the results. Theoretical analysis and simulation results show this novel memristive chaotic system has abundant dynamic behaviors. Based on these abundant dynamical characteristics and its self-excited attractors or hidden attractors under different conditions, this circuit system has the potential application in secure communication and image encryption.

A comparative study of the dynamics of a three-disk dynamo system with and without time delay

The disk dynamo plays an important role in studying the geodynamo and much research works have been devoted to the understanding of dynamo dynamics. This paper further investigates an extended disk dynamo system having three coupled conducting disks and incorporates the interaction-induced time delay in the dynamic governing equations. By carrying out a comparative analysis, the dynamic behaviors of the coupled three-disk dynamo system with and without time delay are studied to explore novel and complex nonlinear dynamic phenomena in the coupled delayed dynamo system. It is found that the double Hopf bifurcations can be induced in the time-delayed dynamo system. Three different topological structures of the unfolding are obtained under different time delays. Accordingly, it is shown that the novel dynamic behaviors, including quasi-periodic torus, three-dimensional torus and the coexistence of multiple attractors, can appear in the time-delayed dynamo system. Furthermore, by performing the continuation analysis on the periodic orbit generated from the Hopf bifurcation of equilibrium, some new coexistence patterns, e.g., the coexistence of periodic orbits and chaos, the coexistence of quasi-periodic orbits and chaos, are observed in the dynamo system with time delay. Based on the obtained results, it is believed that the inclusion of time delay in the modelling of the three-disk dynamo system is necessary and meaningful for developing an in-depth understanding of dynamo dynamics. Finally, the results of theoretical analyses are verified by the numerical simulations.

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.

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.

Cooperation delay induced chaos in an ecological system.

In the present paper, we investigate the impact of time delay during cooperative hunting in a predator-prey model. We consider that cooperative predators do not aggregate in a group instantly, but individuals use different stages and strategies such as tactile, visual, vocal cues, or a suitable combination of these to communicate with each other. We observe that delay in hunting cooperation has stabilizing as well as destabilizing effects in the system. Also, for an increase in the strength of the delay, the system dynamics switch multiple times and eventually become chaotic. We see that depending on the threshold of time delay, the system may restore its original state or may go far away from its original state and unable to recollect its memory. Furthermore, we explore the dynamics of the system in different bi-parameter spaces and observe that for a particular range of other parameter values, the system dynamics switch multiple times with an increase of delay in all the planes. Different kinds of multistability behaviors, the coexistence of multiple attractors, and interesting changes in the basins of attraction of the system are also observed. We infer that depending on the initial population size and the strength of cooperation delay, the populations can exhibit stable coexistence, oscillating coexistence, or extinction of the predator species.

Multistability analysis of a conformable fractional-order chaotic system

In this paper, a new algorithm which is combined conformable fractional derivative and CADM algorithm for solving fractional-order chaotic system is proposed and applied to a new fractional-order Jerk chaotic system. The dissipative and stability of equilibrium points of the system are analyzed. The influences of system parameters and order on the dynamic behaviors of the fractional-order new Jerk chaotic system are analyzed by coexistence of bifurcation diagrams, coexistence of Lyapunov exponent spectrums, attractors coexisting diagrams, attraction basin and dynamics map for initial values. The system exhibits rich dynamic characteristics. Firstly, the system has at least 12 chaotic attractors with different shapes. Multifarious coexistence bifurcation behaviors and coexistence of multiple attractors appear when the traditional parameters and order treated as bifurcation control parameters. More interestingly, attractors coexisting, and multiple state transition under different parameters and orders are analyzed. The result shows that the fractional-order new Jerk chaotic system has rich dynamics characteristics. In addition, the fractional chaotic sequences are generated by DSP, and the results are consistent with numerical simulation. The research results provide guidance for the application and teaching of the conformable fractional-order new Jerk chaotic system.

Global Dynamics and Synchronization in a Duopoly Game with Bounded Rationality and Consumer Surplus

Based on the oligopoly game theory, a dynamic duopoly Cournot model with bounded rationality and consumer surplus is established. On the one hand, the type and the stability of the boundary equilibrium points and the stability conditions of the Nash equilibrium point are discussed in detail. On the other hand, the potential complex dynamics of the system is demonstrated by a set of 2D bifurcation diagrams. It is found that the bifurcation diagrams have beautiful fractal structures when the adjustment speed of production is taken as the bifurcation parameter. And it is verified that the area with scattered points in the 2D bifurcation diagrams is caused by the coexistence of multiple attractors. It is also found that there may be two, three or four coexisting attractors. It is even found the coexistence of Milnor attractor and other attractors. Moreover, the topological structure of the attracting basin and global dynamics of the system are investigated by the noninvertible map theory, using the critical curve and the transverse Lyapunov exponent. It is concluded that two different types of global bifurcations may occur. Because of the symmetry of the system, it can be concluded that the diagonal of the system is an invariant one-dimensional submanifold. And it is controlled by a one-dimensional map which is equivalent to the classical Logistic map. The bifurcation curve of the system on the adjustment speed and the weight of the consumer surplus is obtained based on the properties of the Logistic map. And the synchronization phenomenon along the invariant diagonal is discussed at the end of the paper.

On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition

In this paper, a dynamical two-stage game with R&D competition and joint profit maximization is built. The stability of all the equilibrium points is discussed through Jury condition, and the stability region of the Nash equilibrium point is then given. The influence of the parameters on the system is discussed, and we find that the firm can even benefit from chaos, when it has higher innovation efficiency and higher adjusting speed. And then the coexistence of multiple attractors is studied using basin of attraction. Our research result shows that the coexisting attractors can be observed in the two-parameter bifurcation diagram. At last, the boundary of feasible region, global bifurcations, and formation mechanism of fractal structure of attracting basin are analyzed through critical curves and noninvertible map theory.

Transient responses of nonlinear dynamical systems under colored noise

The influence of colored noise on the transient response of nonlinear dynamical systems is investigated by the generalized cell mapping (GCM) based on the short-time Gaussian approximation (STGA) scheme. The block matrix procedure is introduced into the GCM/STGA method to solve the storage problem caused by the dimensionality of the system. In addition, a parallel calculation strategy can be implemented due to the independence of the storage and the computation of the block matrixes. Taking the well-known Mathieu-Duffing oscillator as an example, the deterministic global properties are first computed with the digraph cell mapping method, in which the coexistence of multiple attractors is observed. Then, the evolutionary processes of transient probability density functions (PDFs) of the response under colored noise from different initial distributions are revealed. Due to the attractive characteristics of the attractor and disturbance generated by colored noise, it can be seen that it takes longer for transient responses from the attractor to achieve the steady-state than the case of the initial distribution which locates around the saddle. The evolutionary processes of transient responses are quite disparate from different initial distributions with the influence of the colored noise, although they both concentrate around the attractor after enough time, which is consistent with the global structure without noise. Furthermore, the effects of the intensity and the correlation time of the colored noise on the mulistability are discussed. The stochastic P-bifurcation occurs with the increase of these two parameters, respectively. The evolutionary directions of the colored-noise–induced bifurcations are opposite in the two cases. Monte Carlo (MC) simulations are in good agreement with the results obtained by the GCM/STGA method based on the block matrix procedure.

Extremely rich dynamical behaviors in a simple nonautonomous Jerk system with generalized nonlinearity : hyperchaos, intermittency, offset-boosting and multistability

This paper investigates the extremely rich dynamical behaviors of the simple Jerk system as proposed by Volos et al. (Nonlinear Dyn 89(2):1047–1061, 2017) based on two main modifications: (i) introduction of a periodic sinusoidal external excitation in the system and (ii) generalization of the nonlinear function of the system in the form $$\varphi _k(x) = 0.5 (\exp (kx)-\exp (-x))$$ as recently proposed by Kengne group (Negou and Kengne in AEU Int J Electron Commun 90:1–19, 2018). These changes are in origin of the observed rich dynamical behaviors including hyperchaos, chaos, intermittency, offset boosting and coexistence of multiple attractors. All these interesting dynamical behaviors are highlighted using the common dynamical tools such as bifurcation diagrams, spectrum of the Lyapunov exponents, two parameters diagrams, phase portraits and Poincare sections. To the best of the author’s knowledge, the addition of an external force in the class of Jerk systems is new and has not been discussed earlier (despite the huge amount of related research works) and thus represents an enriching contribution to the understanding of the dynamics of Jerk’s system. The captured laboratory measurements are in perfect agreement with the theoretical analysis.

Modelling and Analysis of a Host-Parasitoid Impulsive Ecosystem under Resource Limitation

With a long history of theoretical development, biological model has focused on the interaction of a parasitoid and its host. In this paper, two Nicholson-Bailey models with a nonlinear pulse control strategy are proposed and analyzed to examine how limited resource affects the pest control. For a fixed-time discrete impulsive model, the existence and stability of the host-free periodic solution are derived. Threshold analysis suggests that it is critical to release parasitoid in an optimal number in case of the happening of the intra-specific competition, which will seriously affect the pest control. Bifurcation analysis reveals that the model exists complex dynamics including period doubling, chaotic solutions, coexistence of multiple attractors, and so on. For a state-dependent discrete impulsive model, the numerical simulations for bifurcation analysis are studied, the results show that how the key parameters and the initial densities of both populations affect the pest outbreaks, and consequently the relative biological implications with respect to pest control are discussed.

Dynamic analysis of symmetric behavior in flux-controlled memristor circuit based on field programmable gate array

<sec> The lack of the relationship between flux and charge has been made up for by the memristor which is suitable to constructing chaotic circuits as a nonlinear element. Commonly, the memristor-based chaotic systems are constructed by introducing the model of memristor into various classical nonlinear circuits, and more special and abundant dynamic behaviors are existent in these memristive systems. With the deepening of research, several novel nonlinear phenomena of memristor circuits have been found, such as hidden attractors, self-excited attractors and anti-monotonic characteristic. Meanwhile, multistability of a memristor-based circuit explained by the coexistence of multiple attractors with different topological structures is a typical phenomenon in a nonlinear system, and it is also one of the hotspots in this field. In addition, the chaotic sequences generated by the memristive circuits are used as additional signals for information transmission or image encryption. Therefore, the study of modeling memristor systems and analyzing various nonlinear behaviors is of certain valuable.</sec><sec> In this paper, a four-dimensional flux-controlled memeristive circuit is constructed by introducing an active memristor with absolute value into an improved Chua’s circuit, and the special dynamic behaviors are observed. Through the bifurcation diagrams and Lyapunov exponent spectra, the symmetric bifurcations are shown, and the symmetric system states in parameter mappings are found. Besides, the distribution maps of memristive circuit are used to analyze the multistability in a symmetrical attraction domain, and the corresponding phase diagrams are depicted to confirm the existence of multistability. Furthermore, the circuit experiments of the flux-controlled memeristive circuit are implemented by the field programmable gate array simulation, and the experimental results are obtained on a digital oscilloscope, which proves the physical implementability of the memristor-based system.</sec>

A window of multistability in Genesio-Tesi chaotic system, synchronization and application for securing information

In this work, we further investigate the dynamics of the Genesio-Tesi chaotic system which consists of a relatively simple jerk circuit with a quadratic nonlinearity. We complete and enrich the results obtained by Aceng et al. (2016). For this reason, we focus our interest in multistability generation and chaos synchronization as well. By using simulation software tools like PASCAL compiler, Orcad PSPICE and MATLAB, these properties have been characterized via common nonlinear tools including phase portraits, temporal responses, frequency responses, bifurcation diagrams and maximum lyapunov exponent plots. The analysis shows that the Genesio-Tesi system under consideration is able to exhibit complex and interesting behaviors including period doubling bifurcation, chaos, periodic windows and coexistence of multiple attractors. This latter phenomenon has not been found in previous studies of the Genesio-Tesi oscillator thus merits to be shared. We have also shown that Genesio-Tesi systems in their chaotic states can be synchronized and used for a possible masking of information, thus illustrating its importance in engineering. Numerical findings have been validated through experimental studies.