Four-wing Chaotic Attractor Research Articles

Multi-wing chaotic system based on meminductor and its application in image encryption

Meminductor is a novel type of nonlinear device following the memristor, characterized by its memory properties. Currently, research on meminductors is still in its infancy, with their physical devices yet to be formally realized. Therefore, conducting fundamental research on their nonlinear circuit properties and applications is of great significance. In this paper, a new multi-wing chaotic system is proposed based on the mathematical model of a magnetically controlled meminductor. By varying the values of its parameters, the system can generate two-wing, three-wing, and four-wing chaotic attractors. Various analytical methods are employed to study the dynamical behaviours of the proposed chaotic system. The results demonstrate that the system is highly sensitive to its initial conditions and control parameters, which makes it suitable for image encryption. Based on the new system, we propose a new algorithm for image encryption that combines the newly established four-dimensional multi-wing chaotic system with bit plane decomposition technique, firstly, the high four-bit planes containing 94% image information are disordered by S-type permutation, then the disordered bit planes perform operation of XOR with the random matrix generated by chaotic sequences, and finally, the encrypted image is obtained by merging the bit planes.

A novel double-wing chaotic system with infinite equilibria and coexisting rotating attractors: Application to weak signal detection

By introducing a sine function, a non-autonomous multi-wing chaotic system is proposed. The system has an infinite number of equilibrium points and produces symmetrical attractors. The complex dynamical behaviors of the system are demonstrated by phase portraits, Lyapunov exponents spectrum and bifurcation diagram. The effect of driving amplitudes and initial conditions on the resulting system dynamics is then thoroughly investigated. The resulting attractors will enter different oscillatory states or have topological changes. The rotational coexisting attractors depend on the initial conditions and external amplitudes. Besides, a variety of interesting symmetrical transient behaviors and initial-offset boosting behaviors are also found. The driving amplitudes of the system affect the number of attractor wings, a weak signal detection circuit is accordingly designed to estimate the amplitude of periodic weak signals at diverse frequencies. The circuit operates near the switching threshold between the four-wing and two-wing chaotic attractor. Finally, an experimental investigation of the proposed design is performed that demonstrates the theoretical and simulation results.

Four-wing chaotic system and its Hopf bifurcation control

In order to meet the practical needs of Hopf bifurcation control and communication security, we propose a four-wing chaotic attractor and a bifurcation controller based on washout filter. We analyze the dynamic evolution of system by means of bifurcation diagram, Lyapunov exponent and attractor simulation to determine its chaotic characteristics. Based on the high-dimensional bifurcation theory, we discuss the existence of Hopf bifurcation, and design a Hopf bifurcation controller based on washout filter, which is composed of linear and nonlinear terms. Without changing the equilibrium point of system, the controller can generate the Hopf bifurcation of system at the expected position, and realize the control of amplitude and stability of the limit cycle. The simulation results show that the amplitude of the limit cycle and the critical value of bifurcation parameters can be controlled, which verifies the rationality and effectiveness of the controller. This study expands the diversity of Hopf bifurcation control in chaotic system and has potential applications in communication and image encryption.

Multistability Analysis, Coexisting Multiple Attractors, and FPGA Implementation of Yu–Wang Four-Wing Chaotic System

In this paper, we further study the dynamic characteristics of the Yu–Wang chaotic system obtained by Yu and Wang in 2012. The system can show a four-wing chaotic attractor in any direction, including all 3D spaces and 2D planes. For this reason, our interest is focused on multistability generation and chaotic FPGA implementation. The stability analysis, bifurcation diagram, basin of attraction, and Lyapunov exponent spectrum are given as the methods to analyze the dynamic behavior of this system. The analyses show that each system parameter has different coexistence phenomena including coexisting chaotic, coexisting stable node, and coexisting limit cycle. Some remarkable features of the system are that it can generate transient one-wing chaos, transient two-wing chaos, and offset boosting. These phenomena have not been found in previous studies of the Yu–Wang chaotic system, so they are worth sharing. Then, the RK4 algorithm of the Verilog 32-bit floating-point standard format is used to realize the autonomous multistable 4D Yu–Wang chaotic system on FPGA, so that it can be applied in embedded engineering based on chaos. Experiments show that the maximum operating frequency of the Yu–Wang chaotic oscillator designed based on FPGA is 161.212 MHz.

A New 4D Four-Wing Memristive Hyperchaotic System: Dynamical Analysis, Electronic Circuit Design, Shape Synchronization and Secure Communication

In this paper, a simple four-wing chaotic attractor is first proposed by replacing the constant parameters of the Chen system with a periodic piecewise function. Then, a new 4D four-wing memristive hyperchaotic system is presented by adding a flux-controlled memristor with linear memductance into the proposed four-wing Chen system. The memristor mathematical structure model is simple and easy to implement. Dynamical analysis and numerical simulation of the memristive hyperchaotic system are carried out. Then, the electronic circuit of the hyperchaotic system is designed and implemented. The results of numerical simulation are in good agreement with the electronic circuit experiment. In addition, shape synchronization control for the 4D four-wing memristive hyperchaotic system is realized, and a communication system is designed by using the shape synchronization method. Finally, secure signal masking application is implemented on Matlab platform. In the developed secure communication scheme, the information signal overlaps with the chaotic masking signal, which improves the security of the system.

Dynamic Analysis, Circuit Design, and Synchronization of a Novel 6D Memristive Four-Wing Hyperchaotic System with Multiple Coexisting Attractors

In this work, a novel 6D four-wing hyperchaotic system with a line equilibrium based on a flux-controlled memristor model is proposed. The novel system is inspired from an existing 5D four-wing hyperchaotic system introduced by Zarei (2015). Fundamental properties of the novel system are discussed, and its complex behaviors are characterized using phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy. When a suitable set of parameters are chosen, the system exhibits a rich repertoire of dynamic behaviors including double-period bifurcation of the quasiperiod, a single two-wing, and four-wing chaotic attractors. Further analysis of the novel system shows that the multiple coexisting attractors can be observed with different system parameter values and initial values. Moreover, the feasibility of the proposed mathematical model is also presented by using Multisim simulations based on an electronic analog of the model. Finally, the active control method is used to design the appropriate controller to realize the synchronization between the proposed 6D memristive hyperchaotic system and the 6D hyperchaotic Yang system with different structures. The Routh–Hurwitz criterion is used to prove the rationality of the controller, and the feasibility and effectiveness of the proposed synchronization method are proved by numerical simulations.

Modeling of a Chaotic Gyrostat System and Mechanism Analysis of Dynamics Using Force and Energy

Incorporating both viscous friction torque and external torque, a mathematical model of a gyrostat system consisting of three rotors and a fixed outer frame is configured. This model is transformed into a Kolmogorov‐type system for force analysis. The force field in the gyrostat system includes four different torques—inertial, internal, dissipative, and external. Correspondingly, four different energies are identified and the interconversion of energies is analyzed. The Casimir power being equivalent to the error power between the supplied power and the dissipative power is found and used to analyze the mechanism underlying the different dynamical behaviors. A four‐wing chaotic attractor is found when the gyrostat is subject to a combination of inertial, internal, and dissipative torques as well as the addition of an external torque. The bifurcation of the Casimir power and leaps in Casimir energy level are used as indicators of the different definitive dynamics, which is demonstrated to be identical to that appearing in the state bifurcation and the Lyapunov exponent spectrum.

A new method for generating chaotic system with arbitrary shaped distributed attractors.

In this paper, a new method for generating a chaotic system with arbitrary shaped (including heart-shaped, oval, circle, piecewise-linear, and cuboid) distributed attractors is proposed. In this article, a simple four-wing chaotic attractor is first presented by using a periodic piecewise function instead of a constant parameter in the Lorenz system, on the basis of which the chaotic system with arbitrary shaped distributed attractors in the plane can be constructed. This means that the distributed chaotic attractors can be arranged in an arbitrary shape in the plane. The chaotic system can generate any quantity of distributed chaotic attractors, and simulation results show that any desired number of positive Lyapunov exponents can be obtained. Therefore, the chaotic system will have more complicated dynamic characteristics. The dynamical mechanisms of this chaotic system are further investigated, and theoretical analysis and numerical simulation are in accordance with each other, which verifies the effectiveness of the approach. Lastly, the proposed chaotic system is used for image encryption. Numerical results show that the proposed scheme has an excellent performance.

One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics

A novel 3D fractional-order chaotic system is proposed in this paper. And the system equations consist of nine terms including four nonlinearities. It's interesting to see that this new fractional-order chaotic system can generate one-wing, two-wing, three-wing and four-wing attractors by merely varying a single parameter. Moreover, various coexisting attractors with respect to same system parameters and different initial values and the phenomenon of transient chaos are observed in this new system. The complex dynamical properties of the presented fractional-order systems are investigated by means of theoretical analysis and numerical simulations including phase portraits, equilibrium stability, bifurcation diagram and Lyapunov exponents, chaos diagram, and so on. Furthermore, the corresponding implementation circuit is designed. The Multisim simulations and the hardware experimental results are well in accordance with numerical simulations of the same system on the Matlab platform, which verifies the correctness and feasibility of this new fractional-order chaotic system.

Topological horseshoe analysis and field-programmable gate array implementation of a fractional-order four-wing chaotic attractor**Project supported by the National Natural Science Foundation of China (Grant Nos. 61502340 and 61374169), the Application Base and Frontier Technology Research Project of Tianjin, China (Grant No. 15JCYBJC51800), and the South African National Research Foundation Incentive Grants (Grant No. 81705).

We present a fractional-order three-dimensional chaotic system, which can generate four-wing chaotic attractor. Dynamics of the fractional-order system is investigated by numerical simulations. To rigorously verify the chaos properties of this system, the existence of horseshoe in the four-wing attractor is presented. Firstly, a Poincaré section is selected properly, and a first-return Poincaré map is established. Then, a one-dimensional tensile horseshoe is discovered, which verifies the chaos existence of the system in mathematical view. Finally, the fractional-order chaotic attractor is implemented physically with a field-programmable gate array (FPGA) chip, which is useful in further engineering applications of information encryption and secure communications.

Amplitude Control Analysis of a Four-Wing Chaotic Attractor, its Electronic Circuit Designs and Microcontroller-Based Random Number Generator

An exhaustive analysis of a four-wing chaotic system is presented in this paper. It is proved that the evolution range of some variables can be modulated easily by one coefficient of a cross product term. An amplitude-adjustable chaotic circuit is designed, which shows a good agreement with the theoretical analysis. Also, in this paper a microcontroller-based random number generator (RNG) was designed with a nonlinear four-wing chaotic system. RNG studies of the current time have been usually carried out with complicated structures that are costly and difficult to use in real time implementations and that require so much energy consumption. On the other hand, in this paper, as opposed to the disadvantages mentioned here, a microcontroller-based RNG was designed with a four-wing chaotic system (also discussed in the paper) and this was introduced to literature. Microcontroller-based random numbers that passed randomness tests will be available for use in many fields in real life, particularly in encryption.

An enhanced multi-wing fractional-order chaotic system with coexisting attractors and switching hybrid synchronisation with its nonautonomous counterpart

This paper presents a new chaotic system that exhibits a two wing (2W) chaotic attractor in its integer order dynamics, three-wing (3W) and four-wing (4W) chaotic attractors in its fractional-order (FO) dynamics, and an eight-wing (8W) attractor in its nonautonomous fractional dynamics. An interesting feature of the proposed system is that two distinct periodic orbits coexist with a strange attractor that gradually evolves into a 4W attractor. The asymmetry, dissimilarity and topological structure of this proposed system with respect to those available in literature, manifest increased irregularity, which in turn indicate more chaos. Besides, the authors have drawn its comparison with various well-known fractional-order chaotic systems (FOCS)s to prove its enhanced features in terms of higher Lyapunov Exponent, fractional order orbital velocities, bandwidth, density, range of dynamical behaviour, etc. A control scheme is proposed to enable switching hybrid synchronisation between the 8W nonautonomous FOCS and the 4W autonomous FOCS, using the former as master and the latter as slave. This work throws light on the potential practical applicability of the proposed system by designing a circuit using minimum circuit components possible, thus signifying the objectives of the paper are finally achieved.

Generating Four-Wing Hyperchaotic Attractor and Two-Wing, Three-Wing, and Four-Wing Chaotic Attractors in 4D Memristive System

By adding only one smooth flux-controlled memristor into a three-dimensional (3D) pseudo four-wing chaotic system, a new real four-wing hyperchaotic system is constructed in this paper. It is interesting to see that this new memristive chaotic system can generate a four-wing hyperchaotic attractor with a line of equilibria. Moreover, it can generate two-, three- and four-wing chaotic attractors with the variation of a single parameter which denotes the strength of the memristor. At the same time, various coexisting multiple attractors (e.g. three-wing attractors, four-wing attractors and attractors with state transition under the same system parameters) are observed in this system, which means that extreme multistability arises. The complex dynamical behaviors of the proposed system are analyzed by Lyapunov exponents (LEs), phase portraits, Poincaré maps, and time series. An electronic circuit is finally designed to implement the hyperchaotic memristive system.

COMPLEX DYNAMICS OF A SIMPLE 3D AUTONOMOUS CHAOTIC SYSTEM WITH FOUR-WING

The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature[Nonlinear Dyn (2010) 60(3):443-457] with the entitle "A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems" and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.

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.

A fully qualified four-wing four-dimensional autonomous chaotic system and its synchronization

In order to generate a real four-wing chaotic attractor, a new four-dimensional (4D) smooth quadratic autonomous system which is constructed by a linear controller to a three-dimensional (3D) pseudo four-wing system is proposed in this paper. Some complex dynamical behaviors such as equilibrium points and chaos are investigated and analyzed. The corresponding four-wing chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation diagram, analysis of power spectrum and Poincaré maps. Furthermore, an oscillator circuit is designed for implementation, with Multisim observations for verification and demonstration. Finally, an extended synchronization approach is proposed for synchronizing the four-wing chaotic system. On the basis of Jacobin matrix method, it is not necessary to construct the Lyapunov functions, which makes this scheme simpler. Theoretical analysis and numerical simulations demonstrate the validity and feasibility of the proposed control method.

Mechanical analysis of Qi four-wing chaotic system

This paper decomposes vector field of the Qi four-wing chaotic system into four types of torques: inertial torque, internal torque, dissipation and external torque by comparing the system with Kolmogorov system and Euler equation. Angular momentum representing the physical analog of the state variables of the chaotic system is identified. Hamiltonian energy transformation between kinetic energy and potential is exposed using Lie–Poisson bracket. It is discovered that the vector field containing inertial torque and dissipative torque is the basic and necessary condition of producing chaos. Five cases of studies have been conducted to discover the insights and functions of different types of torques of the four-wing chaotic attractor and key factors of producing different types of modes of dynamics.

A novel four-wing non-equilibrium chaotic system and its circuit implementation

In this paper, we construct a novel, 4D smooth autonomous system. Compared to the existing chaotic systems, the most attractive point is that this system does not display any equilibria, but can still exhibit four-wing chaotic attractors. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagram, and Poincare maps. There is little difference between this chaotic system without equilibria and other chaotic systems with equilibria shown by phase portraits and Lyapunov exponents. But the bifurcation diagram shows that the chaotic systems without equilibria do not have characteristics such as pitchfork bifurcation, Hopf bifurcation etc. which are common to the normal chaotic systems. The Poincare maps show that this system is a four-wing chaotic system with more complicated dynamics. Moreover, the physical existence of the four-wing chaotic attractor without equilibria is verified by an electronic circuit.

Simplified hyper-chaotic systems generating multi-wing non-equilibrium attractors

Abstract By a simple feedback control, a new hyper-chaotic system was constructed based on a three-dimensional Lorenz-like chaotic system. This hyper-chaotic system can show butterfly shape (two-wing) strange attractor. The remarkable feature of this system is that it has no equilibrium. By Lyapunov exponents, bifurcation diagram and Poincare maps, the dynamical behaviors of the proposed system were analyzed; and this system was also realized by an electronic circuit. To extend this two-wing non-equilibrium hyper-chaotic system to multi-wing chaotic systems, a piecewise-linear (PWL) function was designed and applied to the proposed hyper-chaotic system, which can produce four-wing, six-wing, eight-wing and ten-wing chaotic attractors. Several experimental circuits were also designed to show these multi-wing chaotic attractors.

Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement

For a three-dimensional autonomous four-wing chaotic attractor, this paper rigorously verifies its chaotic properties by using topological horseshoe theory and numerical calculations. Firstly, an appropriate Poincare section of the chaotic attractor is selected by numerical analysis. Accordingly, a certain first return Poincare map is defined in the Poincare section. Thereafter, by utilizing numerical calculations and topological horseshoe theory, a one-dimensional tensile topological horseshoe in the Poincare section is discovered, which revealed that the four-wing attractor has a positive topological entropy, and verifies the existence of chaos in this four-wing attractor. Finally, by using a FPGA chip, the four-wing chaotic attractor was physically implemented, which is more suitable for engineering applications.