Hidden Chaotic Attractors Research Articles

Autonomous snap oscillator with only one steady state: dynamical probing, controls, pseudo-random number generation and difference synchronization

The theoretical probing, microcontroller implementation, amplitude controls, chaos control, -pseudo-random number generation (PRNG), and difference synchronization of autonomous snap oscillator with only one steady state (ASOOSS) are studied in this paper. The ASOOSS exhibits self-excited complex attractors, periodic oscillations, coexistence of chaotic hidden attractors with a stable steady state, and hidden chaotic attractors. The simulated attractors are endorsed by the microcontroller execution of ASOOSS. Then, the total and partial controls of the amplitude of ASOOSS are demonstrated by using newly inserted parameters. Moreover, the efficacy of the configured single controller in suppressing chaos within ASOOSS is demonstrated through both analytical and numerical analyses. Furthermore, the binary data generated by the ASOOSS-based PRNG successfully passes the NIST 800–22 statistical tests, providing proof of the random nature of the ASOOSS-based PRNG and making it suitable for digital applications based on chaos. Additionally, controllers are devised to enable differential synchronization of three identical coupled chaotic ASOOSS systems. The effectiveness of the differential synchronization approach is validated through numerical simulations of the coupled chaotic ASOOSS systems.

Extreme Homogeneous and Heterogeneous Multistability in a Novel 5D Memristor-Based Chaotic System with Hidden Attractors

This paper proposes a novel five-dimensional (5D) memristor-based chaotic system by introducing a flux-controlled memristor into a 3D chaotic system with two stable equilibrium points, and increases the dimensionality utilizing the state feedback control method. The newly proposed memristor-based chaotic system has line equilibrium points, so the corresponding attractor belongs to a hidden attractor. By using typical nonlinear analysis tools, the complicated dynamical behaviors of the new system are explored, which reveals many interesting phenomena, including extreme homogeneous and heterogeneous multistabilities, hidden transient state and state transition behavior, and offset-boosting control. Meanwhile, the unstable periodic orbits embedded in the hidden chaotic attractor were calculated by the variational method, and the corresponding pruning rules were summarized. Furthermore, the analog and DSP circuit implementation illustrates the flexibility of the proposed memristic system. Finally, the active synchronization of the memristor-based chaotic system was investigated, demonstrating the important engineering application values of the new system.

Chaotic hidden attractor in a fractional order system modeling the interaction between dark matter and dark energy

In this paper the dynamics of a fractional order system modeling the interaction between dark matter and dark energy is analytically and numerically studied. It is shown for the first time that systems modeling the interaction between dark matter and dark energy, chaotic hidden attractors can be present. The chaotic attractor coexists with two asymptotically stable equilibria. Equilibria of the linearized system exhibit a center-like behavior. The numerical integration is done by means of the Adams–Bashforth–Moulton scheme and the finite Lyapunov exponents are numerically determined with a dedicated Matlab code. The 3D representation of the chaotic hidden attractor reveals the fact it is not connected with the equilibria, being “hidden” somewhere in the considered phase space.

A New Memristive System with Extreme Multistability and Hidden Chaotic Attractors and with Application to Image Encryption

Chaotic systems have proven highly beneficial in engineering applications. Pseudo-random numbers produced by chaotic systems have been used for secure communication, notably image encryption. Specific characteristics can increase the chaotic behavior of the system by adding complexity and nonlinearity. The three most well-known characteristics are memristive properties, multistability (coexisting attractors), and hidden attractors. These characteristics strengthen the produced time series’ unpredictability and randomness, strengthening an encryption algorithm’s resistance to many attacks. This study introduces a unique four-dimensional chaotic system with extreme multistability with respect to three initial conditions (including the memristor initial condition) and all previously known properties. It is rare to find an extreme multistable system like this. This system is coupled with a quadratic flux-controlled memristor based on the well-known Sprott J system. This system has a line of unstable equilibrium points with hidden attractors. The memristor displays the characteristic pinched hysteresis loops, where the area inside a loop and the voltage frequency are inversely related. A comprehensive dynamical analysis thoroughly examines all system characteristics and initial conditions. The numerical findings are carefully verified, and an analog circuit is successfully built and simulated. The chaotic sequences generated by this system are combined with deoxyribonucleic acid (DNA) operations and the global bit scrambling (GBS) technique to create an image encryption algorithm that has strong resistance to a variety of potential attacks, including noise, statistical, exhaustive, differential, and cropping attacks.

Fractional-order projection of a chaotic system with hidden attractors and its passivity-based synchronization

This paper presents the fractional-order projection of a chaotic system, which delivers a collection of self-excited and hidden chaotic attractors as a function of a single system parameter. Based on an integer-order chaotic system and the proposed transformation, the fractional-order chaotic system obtains when the divergence of integer and fractional vector fields flows in the same direction. Phase portraits, bifurcation diagrams, and Lyapunov exponents validate the chaos generation. Apart from these results, two passivity-based fractional control laws are designed effectively for the integer and fractional-order chaotic systems. In both cases, the synchronization schemes depend on suitable storage functions given by the fractional Lyapunov theory. Several numerical experiments confirm the proposed approach and agree well with the mathematical deductions.

Construction and implementation of discrete memristive hyperchaotic map with hidden attractors and self-excited attractors

The investigation of discrete memristor and its application in the discrete map is valuable and meaningful. This paper proposes a new one-dimensional map and couples it with a discrete memristor to construct a memristive hyperchaotic map. The most distinctive feature of this map is that it can generate self-excited and hidden chaotic attractor (HCA), the coexistence dynamics of these two types of attractors are different under the conditions of their generations. The relationship between the fixed points and the parameters is analyzed, revealing that it has line fixed points or no fixed points. Moreover, due to the introduction of the memristor, it has a more extensive chaotic range than the original map and can generate hyperchaotic attractors (HA) with better performance metrics. Finally, the map is implemented in a digital circuit to verify the experimental results.

Coexistence of Hidden Attractors in the Smooth Cubic Chua’s Circuit with Two Stable Equilibria

Since the invention of Chua’s circuit, numerous generalizations based on the substitution of the nonlinear function have been reported. One of the generalizations is obtained by substituting cubic nonlinearity for piece-wise linear (PWL) nonlinearity. Although hidden chaotic attractors with a PWL nonlinearity have been discovered in the classical Chua’s circuit, chaotic attractors with a smooth cubic nonlinearity have long been known as self-excited attractors. Through a systematically exhaustive computer search, this paper identifies coexisting hidden attractors in the smooth cubic Chua’s circuit. Either self-excited or coexisting hidden attractors are now possible in the smooth cubic Chua’s circuit with algebraically elegant values of both initial points and system parameters. The newly found coexisting attractors exhibit an inversion symmetry. Both initial points and system parameters are equally required to localize hidden attractors. Basins of attraction of individual equilibria are illustrated and clearly show critical areas of multistability where a tiny drift in an initial point potentially induces jumps among different basins of attraction and coexisting states. Such multistability poses potential threats to engineering applications. The dynamical regions of hidden and self-excited attractors, and areas of stability of equilibria, are illustrated against two parameter spaces. Both illustrations reveal that two nonzero equilibrium points of the smooth cubic Chua’s circuit have a transition from unstable to stable equilibrium points, leading to generations of self-excited and hidden attractors simultaneously.

Hidden multistability of fractional discrete non-equilibrium point memristor based map

At present, the multistability analysis in discrete nonlinear fractional-order systems is a subject that is receiving a lot of attention. In this article, a new discrete non-equilibrium point memristor-based map with γ − th Caputo fractional difference is introduced. In addition, in the context of the commensurate and non-commensurate instances, the nonlinear dynamics of the suggested discrete fractional map, such as its multistability, hidden chaotic attractor, and hidden hyperchaotic attractor, are investigated through several numerical techniques, including Lyapunov exponents, phase attractors, bifurcation diagrams, and the 0 − 1 test. These dynamic behaviors suggest that the fractional discrete memristive map has a hidden multistability. Finally, to validate the presence of chaos, a complexity analysis is carried out using approximation entropy (ApEn) and the C 0 measure. The findings show that the model has a high degree of complexity, which is affected by the system parameters and the fractional values.

Fractal resistive–capacitive–inductive shunted Josephson junction: Theoretical investigation and microcontroller implementation

The paper captures the analytical and numerical investigations of the fractal resistive–capacitive–inductive shunted Josephson junction (FRCLSJJ) and its microcontroller implementation (MCI). The rate equations of FRCLSJJ are established and based on the Routh–Hurwitz criterion, two equilibrium points are reported with one unconditionally stable and the other unstable for the direct current used for junction excitation less than or equal to one. When this current is greater than one, the FRCLSJJ exhibits no equilibrium point. The contribution of fractal parameters to the dynamics of FRCLSJJ is investigated on fast bursting, regular spiking, relaxation behaviors, and periodic bursting. The FRCLSJJ is characterized by fine dynamics such as different presentations of complex behaviors, the coexistence of periodic and chaotic hidden attractors, chaotic hidden attractors, antimonotonicity, periodic attractors, and periodic and chaotic bubble hidden attractors for varying system parameters. And in the last section of the investigation, the MCI of FRCSJJ is realized with the results establishing a qualitative agreement with numerically simulated results.

A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization

The study of hidden attractors plays a very important role in the engineering applications of nonlinear dynamical systems. In this paper, a new three-dimensional (3D) chaotic system is proposed in which hidden attractors and self-excited attractors appear as the parameters change. Meanwhile, asymmetric coexisting attractors are also found as a result of the system symmetry. The complex dynamical behaviors of the proposed system were investigated using various tools, including time-series diagrams, Poincaré first return maps, bifurcation diagrams, and basins of attraction. Moreover, the unstable periodic orbits within a topological length of 3 in the hidden chaotic attractor were calculated systematically by the variational method, which required six letters to establish suitable symbolic dynamics. Furthermore, the practicality of the hidden attractor chaotic system was verified by circuit simulations. Finally, offset boosting control and adaptive synchronization were used to investigate the utility of the proposed chaotic system in engineering applications.

Dynamic Analysis of a Novel 3D Chaotic System with Hidden and Coexisting Attractors: Offset Boosting, Synchronization, and Circuit Realization

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.

Autonomous jerk oscillator with sine nonlinearity and logistic map for sEMG encryption

This research discusses the dynamical behaviors of the autonomous jerk oscillator with sine nonlinearity (AJOSN) and its application to secure s-EMG (Surface ElectroMyoGraphic) data. The AJOSN has two or no-equilibrium points, and the stability of the two equilibrium points indicates that one is stable and the other is unstable. The AJOSN reveals fast-spiking and periodic bursting, relaxing and periodic oscillations, hidden chaotic attractors and coexisting attractors. The numerical analysis results are validated by the Field Programmable Gate Array (FPGA) implementation. Finally, the AJOSN’s chaotic behavior coupled with the logistic map is exploited to encrypt the s-EMG signals. The proposed encryption and decryption method consists of simultaneously encrypting the sEMG signal. The sEMG signal, once transformed into 2D as a grayscale image, undergoes first encryption based on the chaotic signal generated by the AJOSN (Autonomous jerk oscillator with sine nonlinearity) model. The encrypted signal obtained from this first part is called ‘Jerk encrypted signal.’ The second part is to encrypt the ‘Jerk Encrypted Signal’ by the chaotic logistics map. The encryption and decryption results obtained are quite encouraging and offer a great prospect.

A Hidden Chaotic Attractor with an Independent Amplitude-Frequency Controller

A Hidden Chaotic Attractor with an Independent Amplitude-Frequency Controller

Hidden extreme multistability and its control with selection of a desired attractor in a non-autonomous Hopfield neuron

In this article, a single nonautonomous Hopfield neuron is investigated. This neuron involves a new hyperbolic-type memristor as self-synaptic weight. The current–voltage characteristics of the new memristor enables the observation of the hysteresis loop with more than one pinch. The analysis of the equilibria highlighted that the model has hidden dynamics. A plethora of nonlinear phenomena is obtained during the analysis of the model. Among them, the coexistence of an infinite number of symmetric bifurcation diagrams each having its intrinsic energetic singularity. The zooms of the initial condition diagrams enable to observe the coexistence of ten periodic hidden attractors and nine chaotic hidden attractors. The model also exhibited the antimonotonicity phenomenon characterized by the coexistence of an infinite number of bubbles. The control of up to nine coexisting hidden chaotic attractors in a specific domain of the initial condition is successfully realized. This control is achieved by choosing initially an attractor as a designated survivor using the feedback term method. It is then found that the employed method can also be used to control multistability in a system without an equilibrium point. An analog circuit of the model is also designed in PSpice, to further support the results of the theoretical investigations.

Sub-Synchronous Oscillations and Transient Stability of Islanded Microgrid

Sustained sub-synchronous oscillations are observed in an islanded microgrid consisting of grid-forming converters under a large transient disturbance. Such oscillations are studied in terms of various types of hidden attractors, namely, hidden periodic orbit, hidden quasi-periodic orbit, and hidden chaotic attractor, which are shown to coexist with the stable equilibrium point, share different basins of attraction, and be transformed mutually for different parameter values. Moreover, the basins of attraction have fractal boundaries, making the nearby trajectories unpredictable. The transient stability of the islanded microgrid can be improved by eliminating the hidden attractors through adjusting control parameters and triggering a homoclinic bifurcation. Furthermore, transient disturbances can be mitigated by protecting critical converters and accelerating the restarting process. Besides, the emergence of sub-synchronous oscillations can be eliminated through tripping off one or more specific converters. Full-circuit cycle-by-cycle simulations are performed for islanded microgrids comprising i) 5 grid-forming converters, ii) 5 grid-forming converters and a synchronous generator, and iii) 30 grid-forming converters to verify the analytical findings.

New insights to the Hide-Skeldon-Acheson dynamo

<p style='text-indent:20px;'>The present work is devoted to giving new insights into the Hide-Skeldon-Acheson (HSA) dynamo. The paper discusses the locally and globally asymptotical stability of the equilibrium. All orbits of the system are proved to be bounded. The existence of periodic orbits is proved by the generalized Melnikov method. The paper proves rigorously that Hopf bifurcation occurs and gives the formulae to determine the direction, stability and period of bifurcating periodic solutions. Finally, the paper investigates the coexistence of three types of attractors: equilibria, hidden periodic attractors and hidden chaotic attractors.</p>

A Hidden Chaotic Attractor with an Independent Amplitude‐Frequency Controller

In this paper, a three‐dimensional chaotic system with a line equilibrium is studied, in which a single nonbifurcation parameter is used to control the amplitude and frequency. A variety of chaotic signals can be modified using the amplitude‐frequency control switch. The realization of circuit simulation based on multisim further verifies the theoretical analysis. Finally, the method for encrypting color images is tested, and the process performance is valued. It shows that the novel chaotic oscillation has a promising application prospect in image encryption.

Josephson junction model with cosine interference term: Analysis, microcontroller implementation, and network analysis

The dynamical behavior of a single and network of Josephson junction (JJ) model with cosine interference term (CIT) is explored in this paper. The rate-equations describing a single JJ model with CIT have no or two equilibrium points as a function of direct current (DC). The stability of two equilibrium points reveals that one of the equilibrium points is stable node and other is saddle node. The presence of CIT leads to the change of regular spiking and intrinsic bursting to periodic bursting. JJ circuit model with CIT shows the existence of bistable periodic attractors, periodic attractors, hidden chaotic attractors, and coexisting attractors during the numerical simulations. By increasing the coherence parameter, the coexisting attractors are controlled to bistable limit cycles. The microcontroller implementation of the JJ model with CIT is implemented to ascertain the results of numerical simulations. To demonstrate the collective behavior of the JJ model with CIT, a lattice array of identical JJ models with CIT is constructed and studied. By considering the coupling constant as the control parameter, it is demonstrated that the existence of incoherent nodes for lower coupling values and formation of chimera states when the coupling strength is increased.

Microcontroller Implementation, Chaos Control, Synchronization and Antisynchronization of Josephson Junction Model

The microcontroller implementation, chaos control, synchronization, and antisynchronization of the nonlinear resistive-capacitive-inductive shunted Josephson junction (NRCISJJ) model are reported in this paper. The dynamical behavior of the NRCISJJ model is performed using phase portraits, and time series. The numerical simulation results reveal that the NRCISJJ model exhibits different shapes of hidden chaotic attractors by varying the parameters. The existence of different shapes of hidden chaotic attractors is confirmed by microcontroller results obtained from the microcontroller implementation of the NRCISJJ model. It is theoretically demonstrated that the two designed single controllers can suppress the hidden chaotic attractors found in the NRCISJJ model. Finally, the synchronization and antisynchronization of unidirectional coupled NRCISJJ models are studied by using the feedback control method. Thanks to the Routh Hurwitz stability criterion, the controllers are designed in order to control chaos in JJ models and achieved synchronization and antisynchronization between coupled NRCISJJ models. Numerical simulations are shown to clarify and confirm the control, synchronization, and antisynchronization.

An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with “self-excited attractors.” This paper makes a contribution to the topic of fractional-order discrete-time systems with “hidden attractors” by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system.