Multi-wing Attractors Research Articles

A novel grid multiwing chaotic system with only non-hyperbolic equilibria

The structure of the chaotic attractor of a system is mainly determined by the nonlinear functions in system equations. By using a new saw-tooth wave function and a new stair function, a novel complex grid multiwing chaotic system which belongs to non-Shil’nikov chaotic system with non-hyperbolic equilibrium points is proposed in this paper. It is particularly interesting that the complex grid multiwing attractors are generated by increasing the number of non-hyperbolic equilibrium points, which are different from the traditional methods of realising multiwing attractors by adding the index-2 saddle-focus equilibrium points in double-wing chaotic systems. The basic dynamical properties of the new system, such as dissipativity, phase portraits, the stability of the equilibria, the time-domain waveform, power spectrum, bifurcation diagram, Lyapunov exponents, and so on, are investigated by theoretical analysis and numerical simulations. Furthermore, the corresponding electronic circuit is designed and simulated on the Multisim platform. The Multisim simulation results and the hardware experimental results are in good agreement with the numerical simulations of the same system on Matlab platform, which verify the feasibility of this new grid multiwing chaotic system.

Constructing multiwing attractors from a robust chaotic system with non-hyperbolic equilibrium points

ABSTRACTWe investigate a three-dimensional (3D) robust chaotic system which only holds two non-hyperbolic equilibrium points, and finds the complex dynamical behaviour of position modulation beyond amplitude modulation. To extend the application of this chaotic system, we initiate a novel methodology to construct multiwing chaotic attractors by modifying the position and amplitude parameters. Moreover, the signal amplitude, range and distance of the generated multiwings can be easily adjusted by using the control parameters, which enable us to enhance the potential application in chaotic cryptography and secure communication. The effectiveness of the theoretical analyses is confirmed by numerical simulations. Particularly, the multiwing attractor is physically realized by using DSP (digital signal processor) chip.

Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function

In this paper, a simplified memristor-based fractional-order neural network (MFNN) with discontinuous memductance function is proposed. It is essentially a switched system with irregular switching laws and consists of eight fractional-order neural network (FNN) subsystems. The nonlinear dynamics of the simplified MFNN including equilibrium points and their stability, bifurcation and chaos is investigated analytically and numerically. In particular, the effect of the switching jump on the dynamics of the simplified MFNN is explored for the first time. Taking the fractional order, the memristive connection weight or the switching jump as the bifurcation parameter, the dynamics such as chaotic motion, tangent bifurcation and intermittent chaos is identified over a wide range of some specified parameter. It can be seen that the incorporation of the memristors greatly improves and enriches the dynamics of the corresponding FNN. Different from the period-doubling route to chaos, this paper reveals that the mechanism behind the emergence of chaos for the simplified MFNN is the intermittency route to chaos. In particular, for some typical parameter, the existence of chaotic attractors is verified with the phase portraits, bifurcation diagrams, Poincare sections and maximum Lyapunov exponents, respectively. This paper not only provides a way of designing chaotic MFNN with discontinuous memductance function but also suggests a possible method of generating more complicated chaotic attractors, such as multi-scroll or multi-wing attractors.

Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system**Project supported by the National Natural Science Foundation of China (Grant Nos. 51177117 and 51307130).

A novel 5-dimensional (5D) memristive chaotic system is proposed, in which multi-scroll hidden attractors and multi-wing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincaré map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.

Solvability of chaotic fractional systems with 3D four-scroll attractors

One of the questions that has recently predominated the literature is the generation and modulation of strange chaotic attractors, namely the ones with multi scrolls. The fractional theory might be useful in addressing the questions. We use the Caputo fractional derivative together with Haar wavelet numerical scheme to investigate a three-dimensional system that generates chaotic four-wing attractors. Some conditions of stability at the origin (the trivial equilibrium point) are provided for the model. The error analysis shows that the method converges and is concluded thanks to Fubini–Tonelli theorem for non-negative functions and the Mean value theorem for definite integrals. Graphical simulations, performed for some different value of the derivative order α show existence, as expected, of chaotic dynamics characterized by orbits with four scrolls, typical to strange attractors. Hence, fractional calculus appears to be useful in generating and modulating chaotic multi-wing attractors.

Four-wing attractors in a novel chaotic system with hyperbolic sine nonlinearity

Chaotic systems generating multi-wing attractors have received considerable attention in the literature. In this work, we propose a novel three-dimensional chaotic system with hyperbolic sine nonlinearity. It is worth noting that the system is elegant and includes only one parameter. Despite its simple structure, the new system displays double-wing and four-wing chaotic attractors. By studying dynamics of the system, coexistence of limit cycles or chaotic attractors is discovered. The capable of the synchronization of new chaotic system is verified by using an adaptive control. Furthermore, an electronic circuit for implementing the system is reported to indicate its feasibility.

Generating hyperchaotic multi-wing attractor in a 4D memristive circuit

It is very important to generate hyperchaotic attractor with more complicated dynamics for theoretical research and practical application. The paper proposes a novel method to generate hyperchaotic multi-wing attractor. By replacing the resistor in the circuit of modified Lu system with flux-controlled memristor respectively, this new memristive system can exhibit a hyperchaotic multi-wing attractor, and the values of two positive Lyapunov exponents are relatively large. The dynamical behaviors of the proposed system are analyzed by phase portrait, Lyapunov exponents, Poincare maps, and bifurcation diagram. Moreover, the influences of memristor’s strength and position of replaced resistor are analyzed. To further probe the inherent features of the new memristive hyperchaotic system, the circuit implementation is carried out. The proposed method can be easily extended to the generalized Lorenz system family.

A novel methodology for constructing a multi-wing chaotic and hyperchaotic system with a unified step function switching control**Project supported by the National Natural Science Foundation of China (Grant No. 61403143), the Natural Science Foundation of Guangdong Province, China (Grant No. 2014A030313739), the Science and Technology Foundation Program of Guangzhou City, China (Grant No. 201510010124), and the Excellent Doctorial Dissertation Foundation

This paper aims at developing a novel method of constructing a class of multi-wing chaotic and hyperchaotic system by introducing a unified step function. In order to overcome the essential difficulties in iteratively adjusting multiple parameters of conventional multi-parameter control, this paper introduces a unified step function controlled by a single parameter for constructing various multi-wing chaotic and hyperchaotic systems. In particular, to the best of the authors’ knowledge, this is also the first time to find a non-equilibrium multi-wing hyperchaotic system by means of the unified step function control. According to the heteroclinic loop Shilnikov theorem, some properties for multi-wing attractors and its chaos mechanism are further discussed and analyzed. A circuit for multi-wing systems is designed and implemented for demonstration, which verifies the effectiveness of the proposed approach.

Theoretical design approach of four-dimensional piecewise-linear multi-wing hyperchaotic differential dynamic system

This paper reports a novel approach for constructing four-dimensional piecewise-linear multi-wing hyperchaotic differential dynamic system. First, two basic four-dimensional linear systems are obtained. Then, by switching control based on Shilnikov theorem, the double-wing hyperchaotic attractors can be generated. Thirdly, by mth shifting transformation of four-dimensional linear systems, multi-wing hyperchaotic attractors are realized. The designed systems are chaotic in the sense of Smale horseshoe. To confirm the existence of hyperchaotic system, Lyapunov exponent spectrums are further investigated.

Mirror symmetry multi-wing attractors generated from a novel four-dimensional hyperchaotic system

Abstract This research paper introduces mirror symmetry multi-wing attractors from a novel four-dimensional hyperchaotic system. A mirror symmetry conversion-based approach is developed and verified. To confirm the existence of hyperchaotic system, Lyapunov exponent spectrums are investigated. Furthermore, the improved module-based circuit designs of mirror symmetry multi-wing hyperchaotic attractors are further implemented. The proposed novel mirror symmetry multi-wing hyperchaotic attractors are very useful for deliberate generation of chaos in applications.

A Novel No-Equilibrium Chaotic System with Multiwing Butterfly Attractors

Discovering unknown features of no-equilibrium systems with hidden strange attractors is an attractive research topic. This paper presents a novel no-equilibrium chaotic system that is constructed by using a state feedback controller. Interestingly, the new system can exhibit multiwing butterfly attractors. Moreover, a new chaotic system with an infinite number of equilibrium points, which can generate multiscroll attractors, is also proposed by applying the introduced methodology.

Grid multi-wing butterfly chaotic attractors generated from a new 3-D quadratic autonomous system

Due to the dynamic characteristics of the Lorenz system, multi-wing chaotic systems are still confined in the positive half-space and fail to break the threshold limit. In this paper, a new approach for generating complex grid multi-wing attractors that can break the threshold limit via a novel nonlinear modulating function is proposed from the firstly proposed double-wing chaotic system. The proposed method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. The new system is autonomous and can generate various grid multi-wing butterfly chaotic attractors without requiring any external forcing, it also can produce grid multi-wing both on the xz-plane and yz-plane. Basic properties of the new system such as dissipation property, equilibrium, stability, the Lyapunov exponent spectrum and bifurcation diagram are introduced by numerical simulation, theoretical analysis and circuit experiment, which confirm that the multi-wing attractors chaotic system has more rich and complicated chaotic dynamics. Finally, a novel module-based unified circuit is designed which provides some principles and guidelines for future circuitry design and engineering application. The circuit experimental results are consistent with the numerical simulation results.

Simulation studies on the design of optimum PID controllers to suppress chaotic oscillations in a family of Lorenz-like multi-wing attractors

Multi-wing chaotic attractors are highly complex nonlinear dynamical systems with higher number of index-2 equilibrium points. Due to the presence of several equilibrium points, randomness and hence the complexity of the state time series for these multi-wing chaotic systems is much higher than that of the conventional double-wing chaotic attractors. A real-coded genetic algorithm (GA) based global optimization framework has been adopted in this paper as a common template for designing optimum Proportional-Integral-Derivative (PID) controllers in order to control the state trajectories of four different multi-wing chaotic systems among the Lorenz family viz. Lu system, Chen system, Rucklidge (or Shimizu Morioka) system and Sprott-1 system. Robustness of the control scheme for different initial conditions of the multi-wing chaotic systems has also been shown.

A novel four-dimensional multi-wing hyper-chaotic attractor and its application in image encryption

In this paper, a novel four-dimensional chaotic system for generating multi-wing chaotic attractors is proposed, and chaotic and hyper-chaotic attractors are generated in different parameters. Besides, basic dynamical properties of the chaotic system, such as equilibrium point Poincaré mapping, dissipativity, power spectrum, Lyapunov exponent spectrum, bifurcation diagram are studied numerically and theoretically. An analog oscillator circuit is designed for implementing the four-wing hyper-chaotic attractors, and the hardware circuit experimental results are shown to be in good agreement with the numerical simulation results. Finally, the four-wing hyper-chaotic system is used for hybrid image encryption of physical chaos encryption and advanced encryption standard encryption algorithm. Because physical chaos is adopted in this system, there does not exist a definitive relationship between plaintexts and ciphertexts. And the statistical characteristics of ciphertexts should be better than those of any other encryption system.

Two Hyperchaotic Systems with Four-wing Attractors and Their Synchronization

In this paper, an approach to generate multi-wing attractors in coupled chaotic systems is investigated. By coupling two identical Lu systems, novel four-wing hyperchaotic attractors are generated. In particular, novel four-wing hyperchaotic attractors are also generated by coupling a lorenz system and a Lu system. The paper shows that the equilibria of the proposed systems have certain symmetries with respect to specific coordinate planes. In addition, general hybrid projective dislocated synchronization between the two proposed systems is discussed, and this synchronization is applied to secure communication through chaotic masking, using the chaotic signal to mask a continuous signal and a discrete signal. In consideration of interfering random white noise, analyzes the characteristics of noise and then presents a de-noising method using wavelet transform. Simulation results show that the two systems can realize synchronization, further more, the information signal can be recovered undistorted when applying this method to secure communication.

Multi-scroll and multi-wing chaotic attractor generated with Julia process fractal

This paper proposes a general multi-scrolls and multi-wings attractor systems while combining in one hand, Julia’s process with Lorenz’s attractor and in another hand Julia’s process with two coupled Lorenz systems. We have also studied the combination of Julia’s process with the chaotic attractor system proposed in [1].

CONSTRUCTING MULTIWING HYPERCHAOTIC ATTRACTORS

Few reports have introduced chaotic attractors with both multiwing topological structure and hyperchaotic dynamics. A simple construction method, for designing chaotic system with multiwing attractors, is presented in this paper. The number of wings in the attractor was doubled on applying this method to an arbitrary smooth chaotic system. Moreover, the hyperchaotic property is preserved in the new system. A new hyperchaotic system with 16-wing attractors is constructed; the result system is not only verified via numerical simulation but also confirmed by a DSP-based experiment.

DESIGN AND IMPLEMENTATION OF MULTI-WING BUTTERFLY CHAOTIC ATTRACTORS VIA LORENZ-TYPE SYSTEMS

Lorenz system, as the first classical chaotic system, has been intensively investigated over the last four decades. Based on the sawtooth wave function, this paper initiates a novel approach for generating multi-wing butterfly chaotic attractors from the generalized first and second kinds of Lorenz-type systems. Compared with the traditional ring-shaped multi-scroll Lorenz chaotic attractors, the proposed multi-wing butterfly chaotic attractors are much easier to be designed and implemented by analog circuits. The dynamical behaviors of these multi-wing butterfly chaotic systems are further studied. Theoretical analysis shows that every index-2 saddle-focus equilibrium corresponds to a unique wing in the butterfly attractors. Finally, a module-based unified circuit diagram is constructed for realizing various multi-wing butterfly attractors. It should be especially pointed out that this is the first time in the literature that a maximal 10-wing butterfly chaotic attractor is experimentally verified by analog circuits.

A family of hyperchaotic systems with four-wing attractors

It is easy to construct a hyperchaotic system，and many systematic approaches，based on Chua system，can be used for generating multi-scroll attractors. However, few reports discuss the smooth Lorenz-like system with multi-wing attractors，and the attractors with both hyperchaotic dynamics and multi-wing topological structure are seldom reported. Based on the existing smooth hyperchaotic system，the common characteristics of these systems are analyzed. A novel coordination transformation method is used for converting all the attractors' topological structure from two-wing to four-wing. Moreover，the hyperchaotic property is preserved in resulting system，the complexity of attractors is enhanced，and they are more suitable to be used in secure encryption communication.

Scheme for doubling the number of wings in hyperchaotic attractors

The construction methods for designing multi-wing attractors are rarely mentioned little. With the application of coordinate translation, mirror mapping and hysteresis switching to an existing five-dimensional hyperchaotic system, we can double the number of wings and a four-wing attractor is realized. If similar operations have been applied to the system n-1 times, a 2n-wing attractor can be constructed. A simple circuit implementation was designed for doubling the number of wings in attractors. On the basis of preserving the hyperchaotic property of resulting system, the complexity of attractor is enhanced and it is more suitable for use in secure encryption communication.