Double-scroll Chaotic Attractor Research Articles

Cascade tri-neuron hopfield neural network: Dynamical analysis and analog circuit implementation

This paper studies a cascade tri-neuron Hopfield neural network (CTN-HNN) with no connection between the first neuron and the third neuron. Such incompletely connected neuronal structure may be part of complex neural networks with negligible inputs, and may be present in abnormal neural networks with some neurological diseases. It was proved that the stable point, limit cycle, single-scroll chaotic attractor and double-scroll chaotic attractor were observed in the network by adjusting the synaptic weight from the second neuron to itself. To further dynamic analysis, we first demonstrate that this three-dimensional (3D) autonomous nonlinear dynamical system is symmetric about the origin and bounded ultimately, and has one unstable saddle-focus zero point and two symmetric nonzero points with different stability types. Afterwards, taking two synaptic weights as adjustable parameters, the numerical analysis reveals that the CTN-HNN exhibits coexisting chaotic and periodic attractors, coexisting bursting firings and long-term transient chaos by employing multiple numerical methods. Finally, an analog printed circuit is built using one main circuit module and three activation function modules, thus effectively demonstrating the complex dynamics of the network.

Dynamical behaviors, circuit design, and synchronization of a novel symmetric chaotic system with coexisting attractors

In this paper, we introduce a novel three-dimension chaotic system with strange characteristic by applying construction of a 3D chaotic circuit method. Multiple equilibria and abundant coexisting attractors exist in this system. A mathematical model is developed and detailed stability analyses for equilibrium points are executed with obtaining significant results of the period-doubling bifurcation patterns confirmed by phase plane plots and Lyapunov exponent spectra. By varying the initial value and unique controlled parameter, the double-scroll chaotic attractor is broken up into a pair of symmetric singular attractors. Then, the local basins of attraction are investigated concerning the initial condition. Next, the circuit synthesis results generated by Multisim simulation tool validate the self-excitation characteristics of this system. Finally, the feedback control technique is used to study difference synchronization of this system. Main conclusions prove the validity and reliability of difference synchronization.

Dynamic analysis and FPGA implementation of a fourth-order chaotic system with coexisting attractor

To further explore the complex dynamical behaviors in coexisting attractors, a fourth-order chaotic system with four types of coexisting attractors and four unstable equilibrium points is constructed in this paper. The dynamic behavior of the new system is analyzed by means of phase trajectory diagram, time domain waveform diagram, Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. The experimental results show that as the parameters change, the system exhibits rich dynamic behaviors such as stable points, period-doubling bifurcations, coexisting bifurcation modes, and chaotic crises. When the system parameters and memristive parameters change, it is found that the system has different types of coexisting attractors, such as the coexistence of two periodic attractors, the coexistence of two single-scroll chaotic attractors, the coexistence of two double-scroll chaotic attractors, the coexistence of two point attractors. In particular, the system also has the rotation phenomenon of coexisting attractors. Finally, a nonlinear feedback controller is designed, which can make the new system achieve chaos synchronization in a short time. The phase diagram captured by the field-programmable gate array (FPGA) hardware platform is consistent with the numerical simulation results, which proves the feasibility of the system.

Coexisting oscillations and four-scroll chaotic attractors in a pair of coupled memristor-based Duffing oscillators: Theoretical analysis and circuit simulation

We introduce a new four-scroll chaotic system consisting of two autonomous Duffing oscillators with memristive nonlinearity. The memristor emulator consists of a diode bridge and a first-order RC filter. The analytical study of the coupled system shows that it is dissipative, symmetric, and has nine equilibrium points. By using some chaos characterization tools such as bifurcation diagrams, Lyapunov exponent graphs, phase portraits, power spectra, and Poincaré sections, a plethora of dynamical behaviors, such as the period doubling scenario to chaos as well as multistability, are obtained and illustrated. We demonstrate that the system can generate mono-scroll, double-scroll, and four-scroll chaotic attractors according to the coupling parameter. Basins of attraction (illustrating the scenario to four-scroll chaotic attractor) as well as two-parameter Lyapunov diagrams (illustrating the dynamics of the system) are obtained and depicted. To ensure that the new system is implementable, an electronic circuit emulating the model is built and simulated in the PSpice environment.

A four-scroll chaotic generator based on two nonlinear functions and its telecommunications cryptography application

This paper describes a new chaotic jerk system based on two different nonlinear functions. The modified signum and rectangular functions are incorporated into the system at different positions. The system can generate double-scroll and new four-scroll chaotic attractors by adjusting only one coefficient (k). The chaotic behaviors of the proposed system are investigated by the Lyapunov spectrum and period distribution of state variable × of the bifurcations. The complexity of the proposed chaotic system is computed by the Kaplan-Yorke dimension of 2.30. The coexisting attractors and basin of attraction are also included to verify the system's dynamic to the initial conditions. The PCB prototype's commercially available OPAMP and passive components are realized to confirm the numerical studies. The experimental results and the numerical results are agreeable in signals and attractors. The chaotic masking method is applied for encrypting the data for secure communication. The efficient encrypted signal can be obtained, and the decrypted data can be correctly received after successful synchronization.

Dynamical Effects of Offset Terms on a Modified Chua’s Oscillator and Its Circuit Implementation

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.

Analysis of a quarter car suspension based on a Kelvin–Voigt viscoelastic model with fractional-order derivative

The dynamical analysis of a quarter car suspension based on the Kelvin–Voigt​ viscoelastic model with fractional order derivative is analytically and numerically studied in this paper. Depending on the non-dimensional quadratic stiffness parameter, the potential configuration associated with the system under study is possible to obtain monostable and, asymmetric bistable configurations. Thanks to the multiple time scale method, the effects of viscoelastic parameters on the vibration amplitude of the quarter car suspension model are analytically investigated and the numerical simulations confirmed the analytical results. Moreover, the vibration amplitude of the quarter car suspension model can display periodic and dissipative chaotic behaviors by varying the non-dimensional quadratic stiffness, the fractional-order, and the road profile parameters. One scroll and double scroll chaotic attractors are generated by varying the non-dimensional quadratic stiffness parameter. Finally, the threshold conditions for the appearance of homoclinic bifurcation (left well and right well) in the case of asymmetric bistable potential are investigated using Melnikov’s criterion. Melnikov’s predictions are validated numerically by using the basin of attraction of initial conditions. It is found that the decrease of fractional order contributes to increasing the regular motion domain up to a critical value of the fractional-order and after it, decreases rather.

Nonlinear dynamics and chaos in Caputo-like discrete fractional Chen system

In this paper, a fractional Caputo-difference form of discrete Chen system is introduced. The considered system is proposed by implementing the fractional Caputo-like difference operator instead of the standard forward difference operator in the linearized form of the Chen system. The dynamics of the suggested discrete system are numerically illustrated for different fractional orders using phase plots, bifurcation diagrams and Lyapunov exponents. Double scroll chaotic attractors for the considered system are displayed. This study confirms the possible existence of chaos in Caputo-like discrete fractional systems.

A new 2-D multi-stable chaotic attractor and its MultiSim electronic circuit design

<span>A new multi-stable system with a double-scroll chaotic attractor is developed in this paper. Signal plots are simulated using MATLAB and multi-stability is established by showing two different coexisting double-scroll chaotic attractors for different states and same set of parameters. Using integral sliding control, synchronized chaotic attractors are achieved between drive-response chaotic attractors. A MultiSim circuit is designed for the new chaotic attractor, which is useful for practical engineering realizations.</span>

On the dynamics of chaotic circuits based on memristive diode-bridge with variable symmetry: A case study

• This work addresses the dynamics of a generalized memristive diode bridge-based jerk circuit whose I − V characteristic symmetry can be varied. The main motivation of this work lies in the following three keys points: • To evaluate the effects of symmetry-breaking imperfections on the dynamics of the class of memristive circuits considered in this work; • To verify the predictions of the theoretical investigations by performing out laboratory experimental studies of the physical circuit; • To enrich the literature by highlighting new behaviors engendered in symmetric systems in general and the model considered in particular, in the presence of symmetry-breaking. It is widely accepted that for any physical system, symmetries are rarely exact. Therefore, some symmetry imperfections must be always assumed to be present. The dynamics of memristor-based chaotic circuits with symmetric hysteresis loop is well documented. However, only a few works are devoted to the dynamics of these types of circuits when the current-voltage characteristic of the considered memristor is no longer symmetrical. Accordingly, we consider in this work (as a case study) the dynamics of a generalized memristive diode-bridge-based jerk circuit whose symmetry can be varied. We denote by k the dissymmetry coefficient. The tools used for the analysis are the Routh-Hurwitz criterion, bifurcation diagrams, phase portraits, and basins of attraction. It is shown that in the symmetric configuration (i.e. when k = 1.0 ) there are three symmetric equilibria whereas in the asymmetric configuration (i.e. when k ≠ 1.0 ) we always have two equilibrium with fixed position in state space and a third one whose location varies according to the value of the dissymmetry coefficient. The intrinsic nonlinearity of the memristor is responsible for the plethora of nonlinear and complex behaviours observed in both configurations. These include the coexistence of symmetric and asymmetric attractors, coexisting symmetric and asymmetric bubbles of bifurcation, and symmetric and asymmetric double-scroll chaotic attractors, just to name a few. In addition, the experimental investigations agree well with the results of theoretical and numerical studies.

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

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

A Dream that has Come True: Chaos from a Nonlinear Circuit with a Real Memristor

In the last decade, researchers, who work in the field of nonlinear circuits, have the “dream” to use a real memristor, which is the only nonlinear fundamental circuit element, in a new or other reported nonlinear circuit in literature, in order to experimentally investigate chaos. With this intention, for the first time, a well-known nonlinear circuit, in which its nonlinear element has been replaced with a commercially available memristor (KNOWM memristor), is presented in this work. Interesting phenomena concerning chaos theory, such as period-doubling route to chaos, coexisting attractors, one-scroll and double-scroll chaotic attractors are experimentally observed.

Dynamic analysis, FPGA implementation, and cryptographic application of an autonomous 5D chaotic system with offset boosting

An autonomous five-dimensional (5D) system with offset boosting is constructed by modifying the well-known three-dimensional autonomous Liu and Chen system. Equilibrium points of the proposed autonomous 5D system are found and its stability is analyzed. The proposed system includes Hopf bifurcation, periodic attractors, quasi-periodic attractors, a one-scroll chaotic attractor, a double-scroll chaotic attractor, coexisting attractors, the bistability phenomenon, offset boosting with partial amplitude control, reverse period-doubling, and an intermittency route to chaos. Using a field programmable gate array (FPGA), the proposed autonomous 5D system is implemented and the phase portraits are presented to check the numerical simulation results. The chaotic attractors and coexistence of the attractors generated by the FPGA implementation of the proposed system have good qualitative agreement with those found during the numerical simulation. Finally, a sound data encryption and communication system based on the proposed autonomous 5D chaotic system is designed and illustrated through a numerical example.

A chaotic system with a nonlinear term and multiple coexistence attractors

This paper proposes a novel chaotic system characterized by a simple mathematical model and multiple coexisting attractors. Theoretical analysis and numerical simulation show that the proposed system is dissipative, symmetric, and chaotic. The system can generate a double-scroll chaotic attractor with having only one nonlinear term. With changing parameters, a double-scroll attractor is split into two single-scroll attractors. The system has many coexisting attractors, such as one double-scroll chaotic attractor and two single-scroll periodic attractors, two single-scroll chaotic attractors and two single-scroll periodic attractors, and four periodic attractors. The complex dynamic of the system is physically shown by establishing an electronic circuit of the system.

Coexisting attractors, chaos control and synchronization in a self-exciting homopolar dynamo system

PurposeThe purpose of this paper is to investigate coexisting attractors, chaos control and synchronization in a self-exciting homopolar dynamo system in this paper.Design/methodology/approachTwo single controllers are designed and added to the proposed 3D autonomous chaotic system, and its stability at zero equilibrium point is guaranteed by applying an appropriate control signal based on the Lyapunov stability theory.FindingsNumerical simulations reveal that the proposed 3D dynamo system exhibits periodic oscillations, double-scroll chaotic attractors and coexisting attractors. Finally, a single controller is designed for the global asymptotic synchronization of a unidirectionally coupled identical 3D autonomous chaotic system.Originality/valueThe derived results of this paper are new and complement some earlier works. The innovation concludes two points in this paper; coexisting attractors are foundthe and an appropriate control signal based on the Lyapunov stability theory is established. The ideas of this paper can be applied to investigate some other homopolar dynamo systems.

Parallel-Type Asymmetric Memristive Diode-Bridge Emulator and Its Induced Asymmetric Attractor

Symmetry brings beauty, while asymmetry is the general law of nature. This paper reports a novel parallel-type asymmetric memristive diode-bridge (AMDB) emulator, which is implemented by an unbalance diode-bridge linked with a RC filter. Following the voltage constraints of the unbalance diode-bridge, the mathematical model of the parallel-type AMDB emulator is established. Thereafter, the asymmetric property of the hysteresis loops is demonstrated by the numerical simulations and confirmed by the hardware experiments. Furthermore, by importing the parallel-type AMDB emulator into the classical Chua’s circuit, a novel memristive Chua’s circuit is proposed, so that the asymmetric double-scroll chaotic attractor, asymmetric coexisting single-scroll chaotic attractors, and asymmetric coexisting limit cycles can be revealed herein. The parallel-type AMDB emulator enriches the types of memristor emulators and it can mimic the asymmetric property of the physical memristor device.

Infinitely many coexisting attractors of a dual memristive Shinriki oscillator and its FPGA digital implementation

In this paper, a novel chaotic oscillator is proposed, which is derived from the classical Shinriki oscillator by substituting the series-parallel diode loop with a flux-controlled memristor and connecting an active charge-controlled memristor in series with an inductor. The mathematical model of the circuit is established, and the stability distribution maps of three non-zero eigenvalues in the equilibrium plane are obtained. The basic dynamical behaviors depending on the variation of the circuit parameters and memristor initial conditions are investigated by standard nonlinear analysis tools, such as bifurcation diagrams, Lyapunov exponents and phase portraits. Particularly, some striking phenomena, including the routes to double-scroll chaotic attractors, coexisting periodic-chaotic bubbles and asymmetric coexisting behaviors are observed. Furthermore, extreme multistability of the new oscillator is revealed by attraction basins under the initial condition of different dynamic elements. Finally, the Shinriki oscillator with two memristors is realized through Field-Programmable Gate Array (i.e., FPGA) to verify the effectiveness of the numerical simulations.

Polarization-dependent pulse dynamics of mode-locked fiber laser with near-zero net dispersion

We demonstrate novel polarization-dependent pulse dynamics in a fiber laser cavity mode-locked by single-wall carbon nanotubes. By managing the net dispersion and polarization state, we obtain two noise-like pulsing procedures with distinctive polarization characteristics. Ultrafast polarization measurements reveal double pulses are produced by the combined effects of double-scroll chaotic polarization attractors and the polarization mode dispersion of single-mode fiber. The corresponding single-shot spectra provide evidence of coherent energy coupling between pulses during the switching process, due to the polarization hole burning of the active medium. The relative results provide new information for the polarization control of ultrafast lasers.

Analysis and electronic implementation of an absolute memristor autonomous Van der Pol-Duffing circuit

In this paper, a memristor-based chaotic circuit is built by replacing the nonlinear resistor of a unified Van der Pol-Duffing circuit by an ideal and active flux-controlled memristor with an absolute value nonlinearity. The equilibrium points of the mathematical model describing the proposed absolute memristor Van der Pol-Duffing circuit are determined and their stabilities are analyzed thank to the Routh-Hurwitz criteria. The numerical simulations reveal that the proposed absolute memristor circuit exhibits reverse period-doubling to chaos, bistable one-scroll chaotic attractors, double-scroll chaotic attractor, bistable periodic attractors and antimonotonicity phenomenon. Moreover the proposed absolute memristor circuit is implemented in PSIM software package. The results found using the PSIM software package have a good qualitative agreement with those found during the numerical simulations.

Coexistence of attractors in a simple chaotic oscillator with fractional-order-memristor component: analysis, FPGA implementation, chaos control and synchronization

In this paper, the simplest chaotic oscillator with fractional-order-memristor component (SCOF) is proposed. Dynamical characteristics of the proposed chaotic oscillator are investigated both analytically and numerically. The results indicate that the proposed chaotic oscillator possesses novel dynamical characteristics: double-scroll chaotic attractor, four-scroll chaotic attractor and coexisting attractors. The proposed SCOF system is implemented in field programmable gate arrays (FPGA) using the Adomian decomposition method to verify the numerical simulation results. Finally, control and synchronization of the simple chaotic oscillator with fractional-order-memristor component is studied using the linear feedback control and the unidirectional linear error feedback coupling scheme, respectively.