Multi-scroll Systems Research Articles

A general method for generating multi-scroll and multi-wing chaotic systems and its implementation of attractor reproduction

Abstract In comparison to traditional chaotic systems, the multi-scroll and multi-wing chaotic systems are more complicated. The design and execution of sophisticated multi-scroll or multi-wing chaotic attractors attract a lot of attention. However, these constructed nonlinear functions cannot be applied to extended multi-scroll and multi-wing attractors at the same time. To this end, this paper proposes a new function which can be used to generate multi-scroll and multi-wing chaotic attractors in both double-scroll and double-wing chaotic systems. Using this function, multi-scroll and multi-wing chaotic systems can be constructed directly without relying on whether the chaotic system has some symmetry (odd symmetry or even symmetry). The construction method presented is generally applicable to chaotic systems with multi-scroll and multi-wing self-excited attractors.The main point of this method is as follows: firstly, the piecewise linear (PWL) saturation function is nested within the cosine nonlinearity function , and the resulting nested COS-PWL function. Secondly, to enable the expansion of multi-wing and multi-scroll, the nested COS-PWL function is incorporated into the double-wing and double-scroll systems in different manners. The maximum Lyapunov exponent (MLE) and the bifurcation diagram route for increasing the number of wings and scrolls confirm the feasibility and effectiveness of the method. Finally, the three-element method is used to determine a Sinusoidal function, which can generate attractor self-reproduction in the corresponding dimension by replacing the state variables of the multi-scroll and multi-wing systems, so that an infinite number of coexisting attractors can be obtained by simply changing the initial values of the variables, i.e., multiple stability can be generated.

Multi-scroll Systems Synchronization on Strongly Connected Digraphs

In this paper, we study the synchronization problem in complex dynamic networks of Piece Wise Linear (PWL) systems. PWL systems exhibit multi-scrolls and belong to a special class of Unstable Dissipative Systems (UDS). We consider strongly connected digraphs and linear diffusive couplings. The synchronization regions are computed using the concept of disagreement vectors, generalized algebraic connectivity of the network topology, and Lyapunov functions, which provide lower bounds on the coupling gain of the network. Then, different combinations of linear diffusive coupling are explored by changing the observed and measured variables to illustrate the contribution of our results. The theoretical results are validated by numerical simulations.

Designing Twin Memristor-Based Multiscroll Systems by Varying the Flux Variable of Memristor

Memristor, as a nonlinear element in nanometer size, is feasible to generate chaotic signals. Especially, it can improve the randomness of the signals and the complexity of chaotic systems. A novel multiscroll hyperchaotic system based on the flux-controlled memristor is designed. Its twin system with a different topological structure is obtained by varying only the flux variable of the memristor, which is a considerable difference from other chaotic systems existing in the literature. Scroll numbers of the proposed system and its twin, especially, are sensitive to the system parameters. To further investigate the characteristics of the system and its twin, their behaviors with respect to the strengths of the memristor, simulation time and coexistence of multiscroll hyperchaotic attractors are analyzed. Moreover, complex dynamical behaviors of the system and its twin are analyzed by phase diagrams, Poincaré cross-sections, Lyapunov indexes, power spectra, 0-1 test diagrams, and time series. Finally, a novel image encryption scheme is proposed based on the system and its twin. The system and its twin have independent key spaces, and the proposed algorithm makes them have an intersection, which greatly improves the key spaces. In addition, the security and reliability of the image encryption algorithm are demonstrated by different security analysis methods, including correlation analysis, robustness analysis and information entropy test. The results of all experiments prove that the proposed image encryption scheme is superior to other existing ones.

Experimental verification of the multi-scroll chaotic attractors synchronization in PWL arbitrary-order systems using direct coupling and passivity-based control

Because one of the main applications of chaos synchronization is oriented to data encryption and random number generators, fractional-order chaotic systems showing multi-scroll attractors have pointed out as feasible solutions to improve the performance of secure communications schemes. Nowadays, recent works have reported the analysis and control of fractional-order synchronization schemes for multi-scroll chaotic systems. However, the synchronization schemes from those works present mostly theoretical numerical simulations and give circuit simulations employing a simulator software rather than physical implementations. That way, our first contribution is to describe herein the experimental realization of a synchronization scheme for a fractional-order multi-scroll chaotic oscillator with a 4-scroll attractor and fractional-order q=0.9. First, we implemented the multi-scroll systems using an opamp-based analog circuit and an RC-based fractance device to emulate a fractional-order integrator. As a second contribution, we analytically demonstrate the complete synchronization between two identical multi-scroll systems employing two fractional-order controllers and the fractional-order derivative in the Caputo sense. By utilizing the Laplace transform theory, we analyze the one-way coupling method where the response (slave) system’s behavior depends on an identical drive (master) system. As a third contribution, we propose a passivity-based synchronization controller. This synchronization strategy consists of designing a virtual state feedback input and then based on the fractional-order Lyapunov theory, a control law obtains by designing an appropriate storage function to stabilize the system around its stable zero equilibrium. The experimental measures of the implemented synchronization scheme present suitable results compared to analytical and numerical findings.

Scaling of Attractors of a Multiscroll Memristive Chaotic System and its Generalized Synchronization with Sliding Mode Control

Memristor can greatly enhance the complexity of a chaotic system because of its nonlinear characteristics. In this paper, three different memristor models are introduced to the Yang system. The chaotic attractors with single scroll and double scrolls can be obtained by adjusting the action intensities of three memristors and all the attractors inherit the scaling property of attractors of the Yang system. By employing the complex polynomials transformation method in the chaotic system to expand the number of scrolls of the system, the ring-shaped multiscroll attractors are generated, and the number of scrolls can be changed by adjusting the powers of complex polynomials, which show that the memristive system has flexible scalability. Next, a synchronization method for the multiscroll chaotic system is proposed. The generalized synchronization controller and parameter adaptive law are designed by employing sliding mode control. The sufficient conditions for synchronization are given by Lyapunov stability theory. This method can be applied to the synchronization of multiscroll systems generated by means of changing the state variables of the original system by function transformation and then adding the transformation matrix to the system. Compared with the existing synchronization method, this method has a wider scope of application, and it can synchronize two multiscroll chaotic systems with greater difference. In addition, the conditions to be satisfied in this method are simpler. Finally, the method proposed above is applied to the synchronization between a chaotic system with a ring-shaped eight-scroll attractor and a grid-shaped [Formula: see text]-scroll attractor chaotic system with interference signals. The numerical simulation results verify the effectiveness of the method.

On Synchronization of Unidirectionally Coupled Multi-Scroll Systems: Dynamic vs Static Interconnections

Abstract In this work, the occurrence of synchronization in unidirectionally coupled multi-scroll oscillators is investigated. Two types of interconnections are analyzed namely, a traditional static coupling and a second order dynamic coupling, for different input-output combinations. In each case, the stability of the synchronous solution is investigated by using the Hurwitz stability test. The obtained results show that the dynamic coupling scheme has a better performance in the sense that: i) it can induce synchronization even in cases where the static coupling fails, ii) it enlarges the region where the synchronous solution is stable.

Finite-time synchronization of multi-scroll chaotic systems with sigmoid non-linearity and uncertain terms

In this article, the finite-time synchronization of multi-scroll chaotic systems with sigmoid non-linearity in the presence of uncertain stochastic parameters has been studied. The finite-time synchronization is a popular topic from a practical point of view in engineering, which has many applications in the areas like chaos-based communication, image and voice encryptions. The present chaotic system scrolls (up to five) are incorporated with the help of hyperbolic tangent functions (sigmoid non-linearity), but in real applications, there is no limit of multi-scroll in a chaotic system. Finite-time synchronization is achieved with the help of Lyapunov stability using some lemmas and definitions. The Lyapunov exponents are also calculated for multi-scroll chaotic systems with uncertainties and it confirms that the multi-scroll system is chaotic with uncertainties. A drive has been taken to achieve finite-time synchronization of multi-scroll chaotic systems with uncertain parameters, where the multi-scrolls are generated by nonlinear functions.

Multiscroll chaotic system with sigmoid nonlinearity and its fractional order form with synchronization application

In this paper, a multiscroll snap oscillator with hyperbolic tangent function is proposed. There is no limitation in the number of scrolls and it can be increased by proper choice of a specific function. The Lyapunov exponents of the proposed system are obtained to testify the chaotic behavior of the system. Fractional order multiscroll system is derived from its integer order model by using the Adams–Bashforth–Moulton algorithm. A new scheme is applied in order to investigate the synchronization of the multiscroll systems. The main objective of the paper is to propose a multiscroll attractor and show that the number of scrolls can be controlled by the only nonlinear function. Such systems are less investigated in the literatures and has many real time applications like image and voice encryption, random number generators, chaos based communication systems and so on.

Design of fractional-order hyper-chaotic multi-scroll systems based on hysteresis series

This paper develops a novel fractional-order (FO) hyper-chaotic multi-scroll (FOHMS) system derived from a 4-dimensional FO linear system. Hysteresis function series are employed and added to the original system to extend the number of equilibrium points with index 2. The hyper-chaotic dynamics of the new FOHMS system is explored through phase portraits, largest Lyapunov exponents and Poincare maps. A circuit for the 0.96-order FOHMS is implemented and several FOHMS attractors are observed. Numerical and circuit-based simulations show the feasibility of the theoretical formulation.

Generation of a family of fractional order hyper-chaotic multi-scroll attractors

An unified method to yield a family of fractional-order (FO) hyper-chaotic multi-scroll (HCMS) systems in Rn is proposed. Firstly, a new simple 3-dimensional (3-D) FO unstable linear system is introduced. Afterwards, additional variables are added and one nonlinear controller with adjustable parameters is included to generate HCMS attractors. A guideline to construct HCMS systems of any dimension is presented, that is verified along within the dynamics of three examples, namely 4-D, 5-D and 10-D FO HCMS systems. Phase portraits, Poincaré maps and two positive Lyapunov exponents are calculated. Moreover, a circuit of 0.96-order is also designed to realize one 4-D FO HCMS system. Numerical simulations and circuit simulation results show the feasibility of the novel approach.

Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors

In this paper, a new five-dimensional autonomous hyper-chaotic attractor with complex dynamics is introduced. Nonlinear behavior of the proposed system is different with most well-known chaotic and hyper-chaotic attractors and exists interesting aspects in this system. There are eight parameters to control with only one equilibrium point, and the new dynamics can generate four-wing hyper-chaotic and chaotic attractors for some specific parameters and initial conditions. One remarkable feature of the new system is that it can generate double-wing and four-wing smooth chaotic attractors with special appearance. By using the symmetry properties about the coordinates, initial conditions and bifurcation diagrams, many schemes are presented to indicate multi-scroll or multi-wing systems. The phenomenon of multiple attractors is discussed, which means that several attractors created simultaneously from different initial values, and according to this phenomenon, many strange attractors are obtained. Detailed information on the dynamic of the new system is obtained numerically by means of phase portraits, sensitivity to initial conditions, Lyapunov exponents spectrum, bifurcation diagrams and Poincare maps. The stability analysis of the new attractors has been investigated to understand dynamical behaviors nearby equilibrium points.

A family of hyperchaotic multi-scroll attractors in [formula omitted

In this work, we present a mechanism how to yield a family of hyperchaotic multi-scroll systems in Rn based on unstable dissipative systems. This class of systems is constructed by a switching control law changing the equilibrium point of an unstable dissipative system. For each equilibrium point presented in the system a scroll emerges. The switching control law that governs the position of the equilibrium point varies according to the number of scrolls displayed in the attractor. Thus, if two systems display different numbers of scrolls, they have different switching control laws. This paper also presents a generalized theory that explains different approaches such as hysteresis and step functions from a unified viewpoint, extending the concept of chaos in R3 to hyperchaotic multi-scroll systems in Rn,n⩾4. An illustrative example of synchronizing a communication system is given based on the developed theory.

Compound attractors between different chaotic systems

To obtain compound attractors between different chaotic systems, based on theoretical analysis, numerical simulation, and circuit simulation methods, compound attractors between different 2-scroll systems, between different multi-scroll chaotic systems, between 2-scroll system and 2-wing system and between multi-scroll system and multi-wing system, are designed via switching control. Dynamical characteristics of the system are analyzed by observing the attractor phase diagram, the largest Lyapunov exponent and the Poincaré section. A circuit for a compound multiple scroll-multiple wing chaotic attractor is designed and simulated. Numerical simulation and circuit simulation are consistent with each other. It shows that the method of obtaining compound attractors between different chaotic systems via switching control is correct.

Perception-action map learning in controlled multiscroll systems applied to robot navigation

In this paper a new technique for action-oriented perception in robots is presented. The paper starts from exploiting the successful implementation of the basic idea that perceptual states can be embedded into chaotic attractors whose dynamical evolution can be associated with sensorial stimuli. In this way, it can be possible to encode, into the chaotic dynamics, environment-dependent patterns. These have to be suitably linked to an action, executed by the robot, to fulfill an assigned mission. This task is addressed here: the action-oriented perception loop is closed by introducing a simple unsupervised learning stage, implemented via a bio-inspired structure based on the motor map paradigm. In this way, perceptual meanings, useful for solving a given task, can be autonomously learned, based on the environment-dependent patterns embedded into the controlled chaotic dynamics. The presented framework has been tested on a simulated robot and the performance have been successfully compared with other traditional navigation control paradigms. Moreover an implementation of the proposed architecture on a Field Programmable Gate Array is briefly outlined and preliminary experimental results on a roving robot are also reported.

Studies on the construction method of a family of multi-scroll chaotic systems

A method for constructing multi-scroll chaotic systems based on shift transformation is presented. Firstly a proper unstable linear system is selected, and the phase space is divided into several parts by a series of parallel interfaces. Then shift transformations of the linear system are implemented so that each space region is dominated by the corresponding new system. Finally the whole system is obtained using a proper nonlinear function based on signum functions. According to this method new multi-scroll chaotic systems with arbitrary odd number scrolls can be constructed, furthermore, some existing chaotic systems can also be modified into multi-scroll systems. Numerical simulations verify the feasibility of the proposed method.

Reactive navigation through multiscroll systems: from theory to real-time implementation

In this paper a new reactive layer for multi-sensory integration applied to robot navigation is proposed. The new robot navigation technique exploits the use of a chaotic system able to be controlled in real-time towards less complex orbits, like periodic orbits or equilibrium points, considered as perceptive orbits. These are subject to real-time modifications on the basis of environment changes acquired through a distributed sensory system. The strategy is inspired to the olfactory bulb neural activity observed in rabbits subject to external stimuli. The mathematical details of the approach are given including simulation results in a virtual environment. Furthermore the proposed strategy has been tested on an experimental environment consisting of an FPGA-based hardware driving an autonomous roving robot. The obtained results demonstrate the capability to perform a real-time navigation control.

Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control

This paper proposes a saturated function series approach for generating multiscroll chaotic attractors from the fractional differential systems, including one-directional (1-D) n-scroll, two-directional (2-D) nxm-grid scroll, and three-directional (3-D) nxmxl-grid scroll chaotic attractors. Our theoretical analysis shows that all scrolls are located around the equilibria corresponding to the saturated plateaus of the saturated function series on a line in the 1-D case, a plane in the 2-D case, and a three-dimensional space in the 3-D case, respectively. In particular, each saturated plateau corresponds to a unique equilibrium and its unique scroll of the whole attractor. In addition, the number of scrolls is equal to the number of saturated plateaus in the saturated function series. Finally, some underlying dynamical mechanisms are then further investigated for the fractional differential multiscroll systems.

Design and Analysis of Multiscroll Chaotic Attractors From Saturated Function Series

This paper initiates a saturated function series approach for chaos generation. The systematic saturated function series methodology developed here can create multiscroll chaotic attractors from a three-dimensional (3D) linear autonomous system with a simple saturated function series controller, including one-directional n-scroll, two-directional n/spl times/m-grid scroll, and 3-D n/spl times/m/spl times/l-grid scroll chaotic attractors. The dynamical behaviors and chaos generation mechanism of multiscroll systems are further investigated by analyzing the system trajectories. In particular, a two-dimensional (2D) Poincare/spl acute/ return map is rigorously derived for verifying the chaotic behaviors of the double-scroll chaotic attractor, which is a basic generator of various multiscroll chaotic attractors investigated in the paper.