Four-scroll Chaotic Attractor Research Articles

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.

Coexistence of multiscroll chaotic attractors in two coupled inertial Hopfield neurons: numerical simulations and experiment

In this contribution, we consider the dynamics of a pair of coupled inertial neurons with hyperbolic tangent activation function. The two inertial neurons are coupled by adding to each one’s amplitude a perturbation proportional to the other one. The model is governed by a fourth-order autonomous system with hyperbolic tangent nonlinearities. The analysis of the coupled system yields nine equilibrium points some of which experience Hopf type bifurcation. When adjusting the coupling coefficients, striking nonlinear patterns are disclosed such as the coexistence of numerous bifurcation branches, merging crisis, multiple Hopf bifurcations, coexisting self-excited motions (e.g. two coexisting double-scroll chaos, four coexisting period-n cycles, four coexisting single-scroll chaos), and four-scroll chaotic attractors. These latter features are diagnosed with the help of classic numerical tools (e.g. 1D and 2D maximum Lyapunov exponent diagrams, 1D bifurcation diagrams of local peaks of variables, frequency spectrum plots, phase space trajectory plots, and attraction basins). The analogue electronic circuit design of the coupled inertial neurons system is carried out and simulated in PSpice to verify diverse types of features reported during the theoretical study. One of main achievements of the present article is that the coupling of inertial neurons can be regarded as an alternative scheme to obtain multiscroll chaotic signals.

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.

Coupled non-oscillatory Duffing oscillators: Multistability, multiscroll chaos generation and circuit realization

We consider the dynamics of two coupled double-well non-oscillatory cubic Duffing oscillators. The coupling is realized by perturbing each one’s amplitude with a signal proportional to the amplitude of the other one. We prove that the coupling induces in the resulting fourth-order self-driven nonlinear system up to nine equilibrium points and engendered extremely complex dynamic features amongst which the coexistence of several bifurcation branches, multiple Hopf bifurcations (when a single parameter is considered), multistable behaviors (e.g. a pair of coexisting double scroll chaos, four coexisting mono-scroll chaos), and four-scroll chaotic attractor. A detailed investigation of these latter features is carried out by resorting to both analytical (i.e. Hopf bifurcation theorem, Routh–Hurwitz criterion) and numerical tools (e.g. Lyapunov exponent diagrams, 1D bifurcation diagrams, frequency spectrum, phase space trajectory plots, time series and attraction basins). An analog circuit implementing the coupled Duffing oscillators is performed by using simple electronic components (i.e. capacitors, resistors, quadruple operational amplifiers and analog multipliers chips). Experimentally captured phase portraits support the conclusions of the theoretical study. One of the important conclusions of this work is that the coupling of non-oscillatory Duffing type oscillators can be considered as an alternative approach to the generation of multiscroll chaos.

Four-scroll chaotic attractor and four-scroll hyperchaotic attractor generated from a new four-dimensional dynamical system

In this paper, a new 4-D hyperchaotic system with one equilibrium point is first introduced. It contains ten terms with three quadratic nonlinearities. Of particular interest is that this novel system can generate periodic attractor, quasi-periodic attractor, four-scroll chaotic attractor and four-scroll hyperchaotic attractor with the variation of one of its parameters. Major dynamical properties of the new system are investigated such as sensitivity to the initial conditions, dissipativity, equilibrium point stability, Kaplan-Yorke dimension, Lyapunov exponents spectrum and bifurcation diagram. In addition, an equivalent electronic circuit schematic is implemented using Multisim software; the obtained results confirm the feasibility of the proposed system.

Coexisting Three-Scroll and Four-Scroll Chaotic Attractors in a Fractional-Order System by a Three-Scroll Integer-Order Memristive Chaotic System and Chaos Control

Based on the integer-order memristive system that can generate two-scroll, three-scroll, and four-scroll chaotic attractors, in this paper, we found other phenomena that two kinds of three-scroll chaotic attractors coexist in this system with different initial conditions. Furthermore, we proposed a coexisting fractional-order system based on the three-scroll chaotic attractors system, in which the three-scroll or four-scroll chaotic attractors emerged with different fractional-orders q. Meanwhile, with fractional-order q=0.965 and different initial conditions, coexistence of two kinds of three-scroll and four-scroll chaotic attractors is found simultaneously. Finally, we discussed controlling chaos for the fractional-order memristive chaotic system.

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.

Switched-system-based four-scroll chaotic attractor generation and its application in color image encryption

In this paper, a four-scroll chaotic attractor is generated based on a switched system, and moreover, it is applied to color image encryption. First, the canonical Chen system and a modified chaotic system are used as two subsystems to construct the switched system. Meanwhile, a state-dependent switching law with a threshold is designed to regulate the switching between the two subsystems. Then, a four-scroll chaotic attractor is generated by tuning the threshold value. Last, the chaotic sequence extracted from the switched chaotic system is applied to image encryption. The experimental results show that the encryption performance of this switched chaotic system is better than that of a single Chen system.

Coexistence of attractors in integer- and fractional-order three-dimensional autonomous systems with hyperbolic sine nonlinearity: Analysis, circuit design and combination synchronisation

This paper reports the results of the analytical, numerical and analogical analyses of integer- and fractional-order chaotic systems with hyperbolic sine nonlinearity (HSN). By varying a parameter, the integer order of the system displays transcritical bifurcation and new complex shapes of bistable double-scroll chaotic attractors and four-scroll chaotic attractors. The coexistence among four-scroll chaotic attractors, a pair of double-scroll chaotic attractors and a pair of point attractors is also reported for specific parameter values. Numerical results indicate that commensurate and incommensurate fractional orders of the systems display bistable double-scroll chaotic attractors, four-scroll chaotic attractors and coexisting attractors between a pair of double-scroll chaotic attractors and a pair of point attractors. Moreover, the physical existence of chaotic attractors and coexisting attractors found in the integer-order and commensurate fractional-order chaotic systems with HSN is verified using PSIM software. Numerical simulations and PSIM results have a good qualitative agreement. The results obtained in this work have not been reported previously in three-dimensional autonomous system with HSN and thus represent an enriching contribution to the understanding of the dynamics of this class of systems. Finally, combination synchronisation of such three-coupled identical commensurate fractional-order chaotic systems is analysed using the active backstepping method.

An Integer-Order Memristive System with Two- to Four-Scroll Chaotic Attractors and Its Fractional-Order Version with a Coexisting Chaotic Attractor

First, based on a linear passive capacitor C, a linear passive inductor L, an active-charge-controlled memristor, and a fourth-degree polynomial function determined by charge, an integer-order memristive system is suggested. The proposed integer-order memristive system can generate two-scroll, three-scroll, and four-scroll chaotic attractors. The complex dynamics behaviors are investigated numerically. The Lyapunov exponent spectrum with respect to linear passive inductor L and the two-scroll, three-scroll, and four-scroll chaotic attractors are yielded by numerical calculation. Second, based on the integer-order memristive chaotic system with a four-scroll attractor, a fractional-order version memristive system is suggested. The complex dynamics behaviors of its fractional-order version are studied numerically. The largest Lyapunov exponent spectrum with respect to fractional-order p is yielded. The coexisting two kinds of three-scroll chaotic attractors and the coexisting three-scroll and four-scroll chaotic attractors can be found in its fractional-order version.

Dynamics of a new composite four–Scroll chaotic system

A new three-dimensional autonomous chaotic system is proposed. This new system can generate single-scroll, double-scroll, three-scroll and four-scroll attractors under different system parameters. Particularly, it can generate a four-scroll chaotic attractor composed of a large Chua-like attractor and a small Lorenz-like attractor. And the system can also generate a nested three-scroll attractor and the multi-double-scroll chaotic attractor. In addition, the system possesses the chaotic state transition, and the number of scrolls will change in the state transition process. The formation mechanism of the composite four-scroll chaotic attractor is analyzed in detail. The dynamic analysis methods include time series, 0–1 test chart, phase diagram, bifurcation diagram and Lyapunov exponents are used to describe some basic dynamics behaviors of the proposed system.

Memristor-based multi-scroll chaotic system and its pulse synchronization control

The memristor is a nonlinear element and intrinsically possesses memory function. When it works as nonlinear part of a chaotic system, the complexity and the randomness of signal will be enhanced. In this paper memristor is introduced into a three-dimensional chaotic system based on the augmented L system. The interesting and promising behaviors of complex single, double and four-scroll chaotic attractors generated only by varying a parameter have not been reported in memristive chaotic system and thus they deserve to be further investigated. It is also obvious that such a simple change of one parameter could be used to generate a variety of quite complex attractors. Therefore, as a nonlinear device the memristor plays an important role in this system. Firstly, some basic dynamical properties of the memristive chaotic system, including symmetry and in-variance, the existence of attractor, equilibrium, and stability are investigated in detail. By numerically simulating the power spectrum, Lyapunov exponent, Poincare map and bifurcation diagram, in this paper we verify that the proposed system has abundant dynamical behaviors. The sensitivities of system parameters to the chaotic behaviors are further explored by calculating, in detail, its Lyapunov exponent spectrum and bifurcation diagrams. The results of simulation and experiment are in good agreement, thereby proving the veracity of analysis. The memristive chaotic circuit is designed using the memristor, operational amplifier, analog multiplier and other conventional components. The circuit implementation of the memristive system is simulated using SPICE (simulation program with integrated circuit emphasis). The SPICE simulation results and the theoretical analysis are found to be in good agreement, and thus verifying that the system can produce chaos. Pulse synchronization has the following characteristics: low energy consumption, fast synchronization and easy-to-implement single-channel transmission. Therefore, it is more practical in chaotic secure communication. Subsequently the pulse chaos synchronization is realized from the perspective of the maximum Lyapunov exponent, and numerical simulations show the existence of new memristive chaotic system and the feasibility of pulse synchronization control, and also provide an experimental basis for further studying the applications of the memristive chaotic system in voice secure communication and information processing.

A new four-scroll chaotic attractor and its engineering applications

In this paper we investigate a new system of three coupled nonlinear ordinary differential equations, whose dynamics support periodic and chaotic attractors as certain parameters vary. In its most general form, the system has nine parameters. However we can set up to three of these parameters to zero, and still obtain complex dynamics. Here we discuss the case where only two of these parameters are set to zero, and present two-parameter bifurcation linear stability curves for various combinations of the remaining parameters. Then we compute Lyapunov exponents, to verify the regimes of chaotic dynamics, and use adaptive control theory to influence the behaviour. An electronic circuit model of the new chaotic system has been designed and its simulations have been performed using an ORCAD-PSpice program. An experimental realisation of the new chaotic circuit has been carried out and oscilloscope outputs have been compared with numerical (digital) and electronic circuit simulation results. We then used the chaotic system to design a random number generator, and show that the new system has the potential of being used in several scientific and engineering fields such as communication, image processing, physics and mechatronics.

Analysis and control of multiple chaotic attractors from a three-dimensional system

This paper studies a three-dimensional multiple chaotic system with three quadratic nonlinear terms. The system is shown to exhibit multiple periodic attractors and multiple chaotic attractors including a four-scroll chaotic attractor. It is shown analytically and numerically that any neighborhood of the four-scroll chaotic attractor contains repelling sets with positive Lebesgue measures. Moreover, in terms of incommensurate fractional order systems, a simple fractional differentiator-based controller is designed to suppress chaos. Some basic dynamical behaviors of the system are investigated through analytical techniques and numerical simulation.

Hopf Bifurcation and Chaos in Simplest Fractional-Order Memristor-based Electrical Circuit

In this article, we investigate the bifurcation and chaos in a simplest fractional-order memristor-based electrical circuit composed of only three circuit elements: a linear passive capacitor, a linear passive inductor and a non-linear active memristor with two-degree polynomial memristance and a second-order exponent internal state. It is shown that this fractional circuit can exhibit a drastically rich non-linear dynamics such as a Hopf bifurcation, coexistence of two, three and four limit cycles, double-scroll chaotic attractor, four-scroll chaotic attractor, coexistence of one (or two) chaotic attractor with one limit cycle and new chaotic attractor which is not observed in the integer case. Finally, the presence of chaos is confirmed by the application of the recently introduced 0–1 test.

Analysis of a new three-dimensional system with multiple chaotic attractors

In this paper, a new three-dimensional autonomous system with complex dynamical behaviors is reported. This new system has three quadratic nonlinear terms and one constant term. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors in a wide range of system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the new system also can generate four-scroll chaotic attractor. Some basic dynamical behaviors of the system are investigated through theoretical analysis and numerical simulation.

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.

Generating one-, two-, three- and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system

A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincaré maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.

Four-scroll hyperchaos and four-scroll chaos evolved from a novel 4D nonlinear smooth autonomous system

This Letter presents a novel four-dimensional complex nonlinear autonomous system. This system has very rich dynamics and can exhibit hyperchaos, chaos, quasi-periodic and periodic behavior. Applying two symmetrical initial conditions, this system can generate two coexist double-scroll hyperchaotic attractors. More interestingly, this new system can generate not only a four-scroll hyperchaotic but also a four-scroll chaotic attractor. This nonlinear complex system is analyzed by Lyapunov exponent spectrum, bifurcation diagram and Poincaré mapping.

A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors

In this Letter a novel three-dimensional autonomous chaotic system is proposed. Of particular interest is that this novel system can generate two, three and four-scroll chaotic attractors with variation of a single parameter. By applying either analytical or numerical methods, basic properties of the system, such as dynamical behaviors (time history and phase diagrams), Poincaré mapping, bifurcation diagram and Lyapunov exponents are investigated to observe chaotic motions. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.