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On the basis of the chaotic system proposed by Wang et al. in 2023, this paper constructs a 5D fractional-order memristive hyperchaotic system (FOMHS) with multiple coexisting attractors through coupling of magnetic control memristors and dimension expansion. Firstly, the divergence, Kaplan–Yorke dimension, and equilibrium stability of the chaotic model are studied. Subsequently, we explore the construction of the 5D FOMHS, introducing the definitions of the Caputo differential operator and the Riemann–Liouville integral operator and employing the Adomian resolving approach to decompose the linears, the nonlinears, and the constants of the system. The complex dynamic characteristics of the system are analyzed by phase diagrams, Lyapunov exponent spectra, time-domain diagrams, etc. Finally, the hardware circuit of the proposed 5D FOMHS is performed by FPGA, and its randomness is verified using the NIST tool. [ABSTRACT FROM AUTHOR]

In this work, a novel memristive Chua's oscillator circuit is presented. In the proposed circuit, a linear negative resistor, which is parallel coupled with a first-order memristive diode bridge, is used instead of the well-known Chua's diode. Following this, an extensive theoretical and dynamical analysis of the circuit is conducted. This involves numerical computations of the system's phase portraits, bifurcation diagrams, Lyapunov exponents, and continuation diagrams. A comprehensive comparison is made between the numerical simulations and the circuit's simulations performed in Multisim. The analysis reveals a range of intriguing phenomena, including the route to chaos through a period-doubling sequence, antimonotonicity, and coexisting attractors, all of which are corroborated by the circuit's simulation in Multisim. [ABSTRACT FROM AUTHOR]

A new fractional-order cellular neural network (CNN) system is solved using the Adomian decomposition method (ADM) with the hyperbolic tangent activation function in this paper. The equilibrium point is analyzed in this CNN system. The dynamical behaviors are studied as well, using a phase diagram, bifurcation diagram, Lyapunov Exponent spectrum (LEs), and spectral entropy (SE) complexity algorithm. Changing the template parameters and the order values has an impact on the dynamical behaviors. The results indicate that rich dynamical properties exist in the system, such as hyperchaotic attractors, chaotic attractors, asymptotic periodic loops, complex coexisting attractors, and interesting state transition phenomena. In addition, the digital circuit implementation of this fractional-order CNN system is completed on a digital signal processing (DSP) platform, which proves the accuracy of ADM and the physical feasibility of the CNN system. The study in this paper offers a fundamental theory for the fractional-order CNN system as it applies to secure communication and image encryption. [ABSTRACT FROM AUTHOR]

A chaotic oscillator has been designed based on a Sallen Key-type high-pass filter (HPF). The HPF has been converted to a chaotic oscillator using a parallel combination of a PN junction diode as a nonlinear element and an inductor as an energy storage element. The dynamics of the proposed system has been simulated numerically using fourth-order Runge–Kutta method. The circuit exhibits period-doubling route to chaos as well as period-adding route to chaos depending on the choice of system parameters. Striking features like antimonotonicity and coexistence of attractors are also observed. Bifurcation diagram, phase plane plots and spectrum of Lyapunov exponents have been employed to describe the chaotic behavior of the system. A hardware experiment has been carried out to verify the same in the laboratory using off-the-shelf components. [ABSTRACT FROM AUTHOR]

Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of the incommensurate fractional-order Hopfield neural network (FOHNN) system is solved. The equilibrium point of the system is discussed, and the dissipative characteristics are verified and discussed. By varying the order values of the proposed system, different dynamical behaviors of the incommensurate FOHNN system are explored and discussed via bifurcation diagrams, the Lyapunov exponent spectrum, complexity, etc. Finally, using the DSP platform to implement the system, the results are in good agreement with those of the simulation. The actual results indicate that the system shows many complex and interesting phenomena, such as attractor coexistence and an inversion property, with dynamic changes of the order of q0, q1, and q2. These phenomena provide important insights for simulating complex neural system states in pathological conditions and provide the theoretical basis for the later study of incommensurate fractional-order neural network systems. [ABSTRACT FROM AUTHOR]

This paper explores nonlinear analysis of a novel small-scale grid (SSG) and studies the impact of electromagnetic power, random wind power, periodic load and additive white Gaussian noise. Different behaviours such as period doubling bifurcation, chaos, chaos breaking and multistability are investigated and stability issues are revealed in the proposed small-scale grid. Qualitative and quantitative tools such as phase portraits, bifurcation diagrams, Lyapunov exponents, Lyapunov spectrum and basin of attraction are utilised to verify different dynamic behaviours. Erosion of basin region with varying parameter and external noise is reported. Further, the study of unlike behaviours, multistability and coexistence of attractors may be of capital importance in the dynamic evolution of SSG behaviour since serious impediment may occur even after the required safeguards. The present study is expected to be potentially useful in a variety of modern or future power systems, microgrids etc. Numerical simulation is achieved and presented in the MATLAB environment. [ABSTRACT FROM AUTHOR]

In 1971, Chua suggested that there should be a fourth electrical component in addition to resistor, capacitor and inductor. This new component was proposed to be called "memristor". Due to lack of commercially available memristors many emulators are mainly used in nonlinear circuits. Also, the memristance functions are used in many dynamical systems in order to enhance their complexity. Furthermore, the use of memristors in discrete chaotic maps is an interesting research topic, due to the applications of such systems. In this work, a memristor-based Ikeda mapping model is presented by coupling a discrete memristance function with Ikeda map. To investigate system's dynamical behavior a host of nonlinear tools has been used, such as bifurcation and continuation diagrams, maximal Lyapunov exponent diagrams, and Kaplan–Yorke conjecture. Interesting phenomena related to chaos has been observed. More specifically, regular (periodic and quasiperiodic) and chaotic orbits, route to chaos through the mechanism of period doubling and crisis phenomena, have been found. Moreover, higher value of the internal state of the memristor, revealed chaotic behavior in a bigger area. Also, from the comparison of the bifurcation diagrams with the respective continuation diagrams coexisting attractors have been found. Finally, two-parameter bifurcation-like diagrams revealed the system's rich dynamical behavior. • Ikeda map presented chaos through the period doubling and crisis phenomena. • The change in initial conditions revealed the existence of coexisting attractors. • Higher value of the internal state of the memristor, causes chaotic behavior in a bigger area. • Two-parameter bifurcation-like diagrams showed system's dynamical behavior. [ABSTRACT FROM AUTHOR]

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 Cardan's method. Routes to chaos have been investigated through bifurcations structures, Lyapunov exponent and phase portraits. 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), g = 0 with Ω = nω; n a natural number and ... p and q are simple positive integers. Vibrational Resonance (VR), hysteresis and coexistence of several attractors have been studied in details based on the relationship between the frequencies of the AM force. Results of analytical investigations are validated and complemented by numerical simulations. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of bursting, mixed-mode oscillations (MMOs) and horseshoe chaos in a mixed Rayleigh–Liénard oscillator with asymmetric double well potential driven by parametric periodic damping and external excitations. The dynamical behaviours of the considered model, when the exciting frequency is much smaller than the natural frequency, are studied using bifurcation diagrams, Lyapunov exponent diagrams, time histories and phase portraits. It is found that our model displays various bursting and mixed-mode oscillations. Moreover, various mixed-mode oscillation routes to chaos occur in the system. It is also found that the system exhibits two or three coexisting behaviours of attractors when the parametric damping coefficient evolves. On the other hand, the analytical criterion for the appearance of horseshoes chaos is derived using the Melnikov method. A convenient demonstration of the accuracy of the method is obtained from the fractal basin boundary. It is noted that the increase of the impure quadratic damping coefficient, cubic damping coefficient, asymmetric term and the amplitude of the external excitation accentuates the chaotic behaviour of the system. However, the behaviour of the system becomes regular as the parametric damping coefficient increases. [ABSTRACT FROM AUTHOR]

To further understand the dynamical characteristics of chaotic systems with a hidden attractor and coexisting attractors, we investigated the fundamental dynamics of a novel three-dimensional (3D) chaotic system derived by adding a simple constant term to the Yang–Chen system, which includes the bifurcation diagram, Lyapunov exponents spectrum, and basin of attraction, under different parameters. In addition, an offset boosting control method is presented to the state variable, and a numerical simulation of the system is also presented. Furthermore, the unstable cycles embedded in the hidden chaotic attractors are extracted in detail, which shows the effectiveness of the variational method and 1D symbolic dynamics. Finally, the adaptive synchronization of the novel system is successfully designed, and a circuit simulation is implemented to illustrate the flexibility and validity of the numerical results. Theoretical analysis and simulation results indicate that the new system has complex dynamical properties and can be used to facilitate engineering applications. [ABSTRACT FROM AUTHOR]

A new three-dimensional Jerk chaotic system with line equilibrium points is proposed. The system is researched in detail by the Lyapunov exponent graph, bifurcation diagram, phase diagram, and time domain waveform diagram, which show that the system has rich dynamical behaviors, such as eight types of coexisting attractors, extreme multistability of four different attractor states, and offset boosting in two directions. In addition, the system also has six types of transient chaos, which greatly increase the complexity of the system. We study the variation of the spectral entropy (SE) and C0 complexity when the system takes different initial values. Also, in this paper, the initial conditions under which the system is in a synchronized state are determined by initial values with higher complexity. The correctness of the theoretical analysis and numerical simulation is verified by circuit simulation and hardware experiments. Finally, the new system achieves synchronization control utilizing a designed adaptive backstepping controller, laying the foundation for its subsequent use in secure communications. • A new three-dimensional Jerk chaotic system with line equilibrium points is constructed. • The new 3D jerk system has eight types of coexisting attractors and four types of extreme multistability. • The proposed system has six types of transient chaotic behavior. • Adaptive backstepping synchronization avoids the pain and complexity of designing multiple controllers. • Adaptive backstepping synchronization enables synchronization control when system parameters are unknown. [ABSTRACT FROM AUTHOR]

In this paper, a novel four-dimensional autonomous chaotic system based on memristive diode bridge is presented, and it consists of a simple linear oscillation and a memristor. According to the mathematical model of the system, the equilibrium point is analyzed, and the phase portraits, time-domain sequences, bifurcation diagrams and Lyapunov exponent spectrums are numerically simulated. The rich dynamical behavior of proposed system is investigated. It includes multistability, offset boosting and chaotic bursting. In addition, the circuit simulation and Field-Programmable Gate Array (FPGA) hardware experiment are carried out to verify the feasibility of the system. Then the application of proposed system in chaotic image encryption is presented. By carrying out some security performance analyses, we show that the proposed system has good security performance. • A memristor chaotic system with simple circuit structure is proposed. • The memristive chaotic system presented in this paper has rich dynamical behaviors. • The dynamical behavior of offset boosting and multi-stable is proposed. [ABSTRACT FROM AUTHOR]

The dynamics of memristor–based chaotic oscillators with perfect symmetry is very well documented. However, the literature is relatively poor concerning the behaviour of such types of circuits when their symmetry is perturbed. In this paper, we consider the dynamics of a memristive twin-T oscillator. Here, the symmetry is broken by assuming a memristor with an asymmetric pinched hysteresis loop i − v characteristics. A variable disturbance term is introduced into the current-voltage relationship of the memristor in order to obtain an asymmetric characteristic. Phase portraits, bifurcations, basins of attraction, and Lyapunov exponents are used to illustrate various nonlinear patterns experienced by the underlined memristive circuit. It is shown that in the absence of the disturbance term, the i − v characteristic of the memristor is perfectly symmetric which induces typical behaviours such as coexisting symmetric bifurcation and bubbles, spontaneous symmetry-breaking, symmetry recovering, and coexistence of several pairs of mutually symmetric attractors. With the perturbation term, the symmetry of the oscillator is destroyed resulting in more complex nonlinear phenomena such as coexisting asymmetric bubbles of bifurcation, critical transitions, and multiple coexisting (i.e. up to five) asymmetric attractors. Also, PSpice simulation studies confirm well the results of theoretical predictions. [ABSTRACT FROM AUTHOR]

This paper addresses the effects of offset terms on the dynamics of a modified Chua's oscillator. The mathematical model is derived using Kirchhoff's laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Symmetry is an important property shared by a large number of nonlinear dynamical systems. Although the study of nonlinear systems with a symmetry property is very well documented, the literature has no sufficient investigation on the important issues concerning the behavior of such systems when their symmetry is broken or altered. In this work, we introduce a novel autonomous 3D system with cyclic symmetry having a piecewise quadratic nonlinearity φ k (x) = a x − k | x | − x | x | where parameter a is fixed and parameter k controls the symmetry and the nonlinearity of the model. Obviously, for k = 0 the system presents both cyclic and inversion symmetries while the inversion symmetry is explicitly broken for k ≠ 0. We consider in detail the dynamics of the new system for both two regimes of operation by using classical nonlinear analysis tools (e.g. bifurcation diagrams, plots of largest Lyapunov exponents, phase space trajectory plots, etc.). Several nonlinear patterns are reported such as period doubling, periodic windows, parallel bifurcation branches, hysteresis, transient chaos, and the coexistence of multiple attractors of different topologies as well. One of the most gratifying features of the new system introduced in this work is the existence of several parameter ranges for which up to twelve disconnected periodic and chaotic attractors coexist. This latter feature is rarely reported, at least for a simple system like the one discussed in this work. An electronic analog device of the new cyclic system is designed and implemented in PSpice. A very good agreement is observed between PSpice simulation and the theoretical results. [ABSTRACT FROM AUTHOR]

By introducing a discrete memristor and periodic sinusoidal functions, a two-dimensional map with coexisting chaos and hyperchaos is constructed. Various coexisting chaotic and hyperchaotic attractors under different Lyapunov exponents are firstly found in this discrete map, along with which other regimes of coexistence such as coexisting chaos, quasi-periodic oscillation, and discrete periodic points are also captured. The hyperchaotic attractors can be flexibly controlled to be unipolar or bipolar by newly embedded constants meanwhile the amplitude can also be controlled in combination with those coexisting attractors. Based on the nonlinear auto-regressive model with exogenous inputs (NARX) for neural network, the dynamics of the memristive map is well predicted, which provides a potential passage in artificial intelligence-based applications. [ABSTRACT FROM AUTHOR]

Universal, predictive attractor patterns configured by Lyapunov exponents (LEs) as a function of the control parameter are shown to characterize periodic windows in chaos just as in attractors, using a coherent model of the laser with injected signal. One such predictive pattern, the symmetric-like bubble, foretells of an imminent bifurcation. With a slight decrease in the gain parameter, we find the symmetric-like bubble changes to a curved trajectory of two equal LEs in one attractor, while an increase in the gain reverses this process in another attractor. We generalize the power-shift method for accessing coexisting attractors or periodic windows by augmenting the technique with an interim parameter shift that optimizes attractor retrieval. We choose the gain as our parameter to interim shift. When interim gain-shift results are compared with LE patterns for a specific gain, we find critical points on the LE spectra where the attractor is unlikely to survive the gain shift. Noise and lag effects obscure the power shift minimally for large domain attractors. Small domain attractors are less accessible. The power-shift method in conjunction with the interim parameter shift is attractive because it can be experimentally applied without significant or long-lasting modifications to the experimental system. [ABSTRACT FROM AUTHOR]

This paper presents dynamic behavior of a fractional-order memristive time-delay system and its application in image encryption. First, a fractional-order memristive time-delay system is proposed, and the stability and bifurcation behaviors of the system are theoretically analyzed. Some limited conditions for describing the stability interval and switching between different dynamic behaviors are derived. Second, the dynamic characteristics of the system are analyzed through the coexisting attractors, coexisting bifurcation diagrams, the Largest Lyapunov exponents (LLE), the 0-1 test. When parameters change, such as time delay and fractional order, the system transits from steady state to periodic state, single scroll chaotic state, double scroll chaotic state. Furthermore, an image encryption scheme based on the fractional-order memristive time-delay system is introduced, and some statistical features are analyzed. Finally, numerical simulations verify the validity of the theoretical analysis and safety of the image encryption scheme based on the fractional-order delayed memristive chaotic system. [ABSTRACT FROM AUTHOR]

In this paper, a simple 4-dimensional hyperchaotic system is introduced. The proposed system has no equilibria points, so it admits hidden attractor which is an interesting feature of chaotic systems. Another interesting feature of the proposed system is the coexisting of attractors where it shows periodic and chaotic coexisting attractors. After introducing the system, the system is analyzed dynamically using numerical and theoretical techniques. In this analysis, Lyapunov exponents and bifurcation diagrams have been used to investigate chaotic and hyperchaotic nature, the ranges of system parameters for different behaviors and the route for chaos and coexisting attractors regions. In the next part of our work, a synchronization control system for two identical systems is designed. The design procedure uses a combination of simple synergetic control with adaptive updating laws to identify the unknown parameters derived basing on Lyapunov theorem. Microcontroller (MCU) based hardware implementation system is proposed and tested by using MATLAB as a display side. As an application, the designed synchronization system is used as a secure analog communication system. The designed MCU system with MATLAB Simulation is used to validate the designed synchronization and secure communication systems and excellent results have been obtained. [ABSTRACT FROM AUTHOR]

This paper reports the method for constructing multiple coexisting attractors from a chaotic system. First, a new four-dimensional chaotic system with only one equilibrium and two coexisting strange attractors is established. By using bifurcation diagrams and Lyapunov exponents, the dynamical evolution of the new system is presented. Second, a feasible and effective method is applied to construct an infinite number of coexisting attractors from the new system. The core of this method is to batch replicate the attractor of the system in phase space via generating multiple invariant sets and the generation of invariant sets depends on the equilibria, which can be extended by using some simple functions with multiple zeros. Finally, we give some numerical results of the appearance of multiple coexisting attractors in the system with sine and sign functions for demonstrating the effectiveness of the method.

This paper generates an augmented hyperchaotic system from the famous Lorenz system. The hyperchaotic system has complex dynamic properties, including stability, periodicity, multiple coexisting attractors, period-doubling and Hopf bifurcations, and hyperchaos for different parameter conditions and all these dynamic properties are presented by detailed theoretical and numerical analysis. Moreover, the finite-time synchronization of the hyperchaotic system is considered by using the state-error controller. Both the sufficient conditions for finite-time synchronization and the corresponding finite time are strictly established.

In this paper, based on the Adomian decomposition method (ADM) semi analytical solution algorithm, dynamics and complexity of the conformable fractional‐order two‐machine interconnected power system are investigated numerically by the bifurcation diagram, Lyapunov exponents (LEs), chaos diagram, and modified multiscale sample entropy (MM‐SampEn) algorithm separately. The results show that the system has rich dynamics. The angular instability is found and its frequency of occurrence can be judged by the number of scrolls of the chaotic attractor. The coexisting attractors are observed by changing the initial value and the reason is discussed. The high‐complexity region is determined, and MM‐SampEn complexity can indicate different coexisting attractors of the system. The research results in this paper lay a theoretical basis for the application of the conformable fractional‐order two‐machine interconnected power system. [ABSTRACT FROM AUTHOR]

This paper investigates multistability in a 3D autonomous system with different types of chaotic attractors, which are not in the sense of Shil'nikov criteria. First, under some conditions, the system has infinitely many isolated equilibria. Moreover, all equilibria are nonhyperbolic and give the first Lyapunov coefficient. Furthermore, when all equilibria are weak saddle-foci, the system also has infinitely many chaotic attractors. Besides, the Lyapunov exponents spectrum and bifurcation diagram are given. Second, under another condition, all the equilibria constitute a curve and there exist infinitely many singular degenerated heteroclinic orbits. At the same time, the system can show infinitely many chaotic attractors. [ABSTRACT FROM AUTHOR]

Summary: The analysis, electronic validation and chaos control of nonsmooth‐air‐gap brushless direct current motor (BLDCM) running under no loading conditions are investigated in this article. The nonsmooth‐air‐gap BLDCM is described by a system of three‐dimensional autonomous equations. The stability of equilibrium points found is studied. Different dynamical behaviors of nonsmooth‐air‐gap BLDCM including periodic and chaotic spiking oscillations, monostable and bistable double‐scroll chaotic attractors and coexisting attractors are revealed using numerical methods such as two dimensional largest Lyapunov exponents (LLEs) and isospike graphs associated with two parameters of nonsmooth‐air‐gap BLDCM. Moreover, an analog circuit is designed and implemented in OrCAD‐PSpice software to confirm the dynamical behaviors found in nonsmooth‐air‐gap BLDCM during the numerical simulations. Finally, a simple and single controller is designed and added to the chaotic nonsmooth‐air‐gap BLDCM in order to suppress chaotic behavior. The performance of the proposed simple and single controller is illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

This article presents a hyperchaotic system of five-dimensional autonomous ODEs that has five cross-product nonlinearities. Under certain parametric conditions, it exhibits three different types of hyperchaotic and chaotic systems which correspond to six hyperchaotic attractors with a non-hyperbolic equilibrium line, four chaotic attractors with seventeen hyperbolic equilibria, and four chaotic attractors with only one hyperbolic equilibrium, respectively. The fundamental dynamics are analyzed theoretically and numerically, such as the onset of hyperchaos and chaos, routes to chaos, persistence of chaos, coexistence of attractors, periodic windows and bifurcations. It is particularly shown that the coexisting attractors of the 5D system inside the hypercone are symmetric. Some dynamical characteristics of these attractors are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate coexisting strange nonchaotic attractors (SNAs) in a quasiperiodically forced system. We also describe the basins of attraction for coexisting attractors and identify the mechanism for the creation of coexisting attractors. We find three types of routes to coexisting SNAs, including intermittent route, Heagy–Hammel route and fractalization route. The mechanisms for the creation of coexisting SNAs are investigated by the interruption of coexisting torus-doubling bifurcations. We characterize SNAs by the largest Lyapunov exponents, phase sensitivity exponents and power spectrum. Besides, the SNAs with extremely fractal basins exhibit sensitive dependence on the initial condition for some particular parameters. [ABSTRACT FROM AUTHOR]

A new 4D chaotic system with infinitely many equilibria is proposed using a linear state feedback controller in the Sprott C system. Although the new 4D chaotic system has only two nonlinear terms, it has rich dynamic characteristics, such as hidden attractors and coexisting attractors. Besides, the freedom of offset boosting of a variable is achieved by adjusting a controlled constant. The dynamic characteristics of the new chaotic system are fully analyzed from the aspects of phase portraits, bifurcation diagrams, Lyapunov exponents and Poincaré maps. The corresponding analogue electronic circuit is designed and implemented to verify the new 4D chaotic system. By taking advantage of the complex dynamic properties of the new chaotic system, a random number generator algorithm is proposed. [ABSTRACT FROM AUTHOR]

Although multiwing hidden attractor chaotic systems have attracted a lot of interest, the currently reported multiwing hidden attractor chaotic systems are either with no equilibrium point or with an infinite number of equilibrium points. The multiwing hidden attractor chaotic systems with stable equilibrium points have not been reported. This paper reports a four-wing hidden attractor chaotic system, which has only one stable node-focus equilibrium point. The novel system can also generate a hidden attractor with one-wing and hidden attractors with quasi-periodic and periodic coexistence. In addition, a self-excited attractor with one-wing can be generated by adjusting the parameters of the novel system. The hidden attractors of the novel system are verified by the cross-section of attraction basins. And the hidden behavior is investigated by choosing different initial states. Moreover, the coexisting transient four-wing phenomenon of the self-excited one-wing attractor system is studied by the time domain waveforms and attraction basin. The dynamical characteristics of the novel system are studied by Lyapunov exponents spectrum, bifurcation diagram and Poincaré map. Furthermore, the novel hidden attractor system with four-wing and one-wing are implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Chongqing University of Posts & Telecommunications (Natural Science Edition) is the property of Chongqing University of Posts & Telecommunications and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Universal, predictive attractor patterns configured by Lyapunov exponents (LEs) as a function of the control parameter are shown to characterize periodic windows in chaos just as in attractors, using a coherent model of the laser with injected signal. One such predictive pattern, the symmetric-like bubble, foretells of an imminent bifurcation. With a slight decrease in the gain parameter, we find the symmetric-like bubble changes to a curved trajectory of two equal LEs in one attractor, while an increase in the gain reverses this process in another attractor. We generalize the power-shift method for accessing coexisting attractors or periodic windows by augmenting the technique with an interim parameter shift that optimizes attractor retrieval. We choose the gain as our parameter to interim shift. When interim gain-shift results are compared with LE patterns for a specific gain, we find critical points on the LE spectra where the attractor is unlikely to survive the gain shift. Noise and lag effects obscure the power shift minimally for large domain attractors. Small domain attractors are less accessible. The power-shift method in conjunction with the interim parameter shift is attractive because it can be experimentally applied without significant or long-lasting modifications to the experimental system.

In recent years designing new multistable chaotic oscillators has been of noticeable interest. A multistable system is a double-edged sword which can have many benefits in some applications while in some other situations they can be even dangerous. In this paper, we introduce a new multistable two-dimensional oscillator. The forced version of this new oscillator can exhibit chaotic solutions which makes it much more exciting. Also, another scarce feature of this system is the complex basins of attraction for the infinite coexisting attractors. Some initial conditions can escape the whirlpools of nearby attractors and settle down in faraway destinations. The dynamical properties of this new system are investigated by the help of equilibria analysis, bifurcation diagram, Lyapunov exponents' spectrum, and the plot of basins of attraction. The feasibility of the proposed system is also verified through circuit implementation. [ABSTRACT FROM AUTHOR]

In this study, another one-dimensional sinusoidal chaotic map around the bisector is constructed, which consists of two sine functions with irrational frequency ratios but comparable amplitude and phase. This design is motivated by the process equation. Investigating the unusual dynamics of this map and contrasting them with the process equation are the goals of this work. When two sine functions are included in a map, their interaction results in complex behaviors that have not been seen for the process equation. Like the process equation, this newly created map has an endless number of fixed points, but unlike the process equation, their stability cannot be fully defined. Furthermore, the suggested chaotic map can display a biotic-like time series when escaping regions are produced; however, unlike the process equation, these biotic-like time series only consist of transient components. In other words, the introduced map converges to a fixed point or periodic solution and never becomes unbounded because of the unlimited number of stable and unstable fixed points. The parameters of the map determine how long these biotic-like transient sections last. Additionally, coexisting attractors and multi-stability are seen in this map, much like in the process equation. The significant dynamics of this map are examined with time series, cobweb, bifurcation, and Lyapunov exponent diagrams. Tests are conducted before and after the creation of the escaping regions. Additionally, two-dimensional bifurcation diagrams are used to study the simultaneous impact of many pairs of factors on the map's dynamics. [ABSTRACT FROM AUTHOR]

Copyright of Control Theory & Applications / Kongzhi Lilun Yu Yinyong is the property of Editorial Department of Control Theory & Applications and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper investigates the dynamical behavior of the Tamasevicius et al. (1997) oscillator (named TCMNL hereafter) considering memristor as the nonlinear element by replacing the single diode in the original circuit. Various methods for detecting chaos/hyperchaos including bifurcation diagrams, spectrum of Lyapunov exponent, two parameter Lyapunov exponent, Poincaré sections and phase portraits are exploited to establish the connection between the system parameters and various complicated dynamics. By tuning the system parameters, some striking phenomena such as quasi-periodic oscillations and asymmetric pair of stable/unstable attractors are depicted. It is also found that the considered memristor induces the phenomenon of coexistence of attractors in wide ranges of bifurcation parameter. Finally, the hardware circuit is implemented and experimental observations are found to be in good agreement with the numerical investigations. [ABSTRACT FROM AUTHOR]

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the system. [ABSTRACT FROM AUTHOR]

The coexisting attractors which means multiple attractors with independent domains of attraction yield simultaneously in a system is of recent interest. In this paper, we propose a new 3D autonomous quadratic chaotic system as a typical example with the presence of the coexisting attractors. Some basic dynamical properties of the system are presented. The existence of the Hopf bifurcation is established by analyzing the corresponding characteristic equation. It shows that the system coexists double Hopf bifurcation at equilibria as the parameter passes a critical value. The coexisting point, periodic, chaotic attractors in the system are numerically investigated by bifurcation diagrams, Lyapunov exponents and phase diagrams. [ABSTRACT FROM AUTHOR]

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 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]

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]