Multi-wing Chaotic Attractors Research Articles

Generating Multiwing Hidden Chaotic Attractors With Only Stable Node-Foci: Analysis, Implementation, and Application

Based on Shil'nikov criterion, i.e., extending the number of unstable saddle-focus -type equilibria of chaotic system, a rich and varied multiwing chaos with complicated topology and remarkable pseudorandomness has been coined. However, it is a challenging task to construct multiwing chaos with only stable node-focus -type equilibria as the Shil'nikov criterion is not applicable in the circumstances. To eliminate this difficulty and enrich the multiwing chaos, we construct a simple three-dimensional autonomous system with only two symmetric stable equilibria, and further introduce a sawtooth wave modulation function to configure more equilibria. The highlight is that multiwing chaotic attractors can be generated from system with two or multiple stable node-foci. This characteristic indicates that the attractors can be hidden oscillating and are difficult to be located in the small neighborhoods of equilibria, which makes the system has good concealment performance for security applications. The design mechanism and system properties are theoretically analyzed and numerically investigated using phase portraits, finite-time Lyapunov exponents, and basins of attraction. Hardware experiments based on field-programmable gate array verify the feasibility of the system. Finally, based on the generated multiwing hidden chaotic attractors, an image encryption scheme with high security performance is designed to enhance its application.

Finite-time synchronization of fractional multi-wing chaotic system

The objective of this article is to obtain multi-wing chaotic attractors of fractional chaotic systems through computerized symbolic computation. By applying the Julia fractal technique, the different number wing attractors are constructed for proposed equations. Moreover, the dynamics of the multi-wing system are analyzed by phase diagram, Poincare mapping, etc Consequently, the system exhibits complex dynamics, and the motion states at different order can be known from the bifurcation diagram with the change of order. Additionally, aiming at multi- wing fractional chaotic system, the controllers are designed, and the finite time synchronization control of the proposed system is performed. The results prove that the proposed finite-time synchronization method has important research value in the field of engineering.

A Novel Memristor-Based Multi-Vortex Hyper-Chaotic Circuit Design and its Application in Image Encryption

This paper proposes a new four-dimensional hyper-chaotic system capable of generating multi-wing chaotic attractors by introducing active magnetron memristors, multi-segmented square functions and trigonometric functions. The dynamical properties of this new hyper-chaotic system, such as equilibrium point, dissipation, Lyapunov exponential spectrum, bifurcation diagram and Poincaré cross-section and attraction basin, are analyzed theoretically and simulated numerically, and the complexity of this system with different parameters is analyzed. It is observed that this hyper-chaotic system has periodic, chaotic and hyper-chaotic variations with an infinite number of equilibria and coexisting attractors under different parameter conditions. The circuit simulation was performed using Multisim and the results obtained were consistent with the numerical analysis of the dynamics, and the chaotic circuit system is designed by FPGA to verify the realizability of the system. Finally, an image encryption algorithm is designed in conjunction with the DNA algorithm to enable a new system chaotic sequence for image encryption. The results show that the hyper-chaotic system has rich dynamical behavior and has high-security performance when applied to image encryption with strong chaotic key and plaintext sensitivity and large key space in image encryption.

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-directional annular multi-wing chaotic system based on Julia fractals

Fractals and chaos are inextricably linked, and relevant study has been done on both. However, the studies that combine fractals and chaotic systems to produce multi- scroll or multi-wing chaotic attractors are all extend multi-scroll or multi-wing in the even direction. To this end, by improving the traditional fractal mapping, a chaotic system that can extend multiple wings in any direction of odd and even is designed and analyzed. The main point of this method is as follows: Firstly, a new family of even-symmetric segmented composite functions is proposed and applied to a family of generalized Lorenz systems, which can generate multi-wing chaotic systems in two directions and the number of wings in each direction can also be increased with the parameter N. Secondly, the improved Julia fractal is applied to two-direction multi-wing chaotic systems to generate a class of single-direction multi-wing chaotic systems. Finally, the combination of this class of single-direction multi-wing chaotic systems and higher-order Julia fractals is used to realize chaotic systems with extended multi-wing in odd and even arbitrary directions. Computer simulations confirm the method's viability. In addition, the maximum Lyapunov exponent, bifurcation diagram, Poincare diagram, and PE complexity chaos diagram are also analyzed as examples of multi-directional multi-wing Lü chaotic systems, demonstrating the system's superiority in odd directions over even directions and the existence and effective scalability of multi-directional multi-wing chaotic systems.

A new multi-wing chaotic attractor with unusual variation in the number of wings

A new multi-wing chaotic system with some unusual properties is reported in this paper. Here, the number of wings and the amplitude of the chaotic attractor depend on (i) simulation time, (ii) initial conditions, and (iii) system parameters. The amplitude and number of wings of the multi-wing chaotic attractor are not reaching to a fixed chaotic attractor, as in the case of other chaotic systems like Lorenz, even after a very large simulation time of 800000 s. On the other hand, the Lyapunov exponents of the system remain the same with an increasing trend of the simulation time. Moreover, the amplitude and the number of wings of the chaotic attractor keep varying with the simulation time when the initial conditions and system parameters change. Such an unusual property of a chaotic attractor is termed as dynamic-chaotic attractor. In addition to these behaviors, different complex properties of this new chaotic system are explored by plotting phase portraits, Lyapunov spectrum, bifurcation diagrams and Poincare maps. Both homogeneous and heterogeneous multi-stability are found; the co-existence of 7 multi-wing chaotic attractors is reported. Hence, this kind of system is rare in the literature.

Design of Grid Multi-Wing Chaotic Attractors Based on Fractional-Order Differential Systems

In this article, a new method for generating grid multi-wing chaotic attractors from fractional-order linear differential systems is proposed. In order to generate grid multi-wing attractors, we extend the method of constructing heteroclinic loops from classical differential equations to fractional-order differential equations. Firstly, two basic fractional-order linear systems are obtained by linearization at two symmetric equilibrium points of the fractional-order Rucklidge system. Then a heteroclinic loop is constructed and all equilibrium points of the two basic fractional-order linear systems are connected by saturation function switching control. Secondly, the theoretical methods of switching control and construction of heteromorphic rings of fractal-order two-wing and multi-wing chaotic attractors are studied. Finally, the feasibility of the proposed method is verified by numerical simulation.

Hidden multiwing chaotic attractors with multiple stable equilibrium points

PurposeThe purpose of this paper is to construct a multiwing chaotic system that has hidden attractors with multiple stable equilibrium points. Because the multiwing hidden attractors chaotic systems are safer and have more dynamic behaviors, it is necessary to construct such a system to meet the needs of developing engineering.Design/methodology/approachBy introducing a multilevel pulse function into a three-dimensional chaotic system with two stable node–foci equilibrium points, a hidden multiwing attractor with multiple stable equilibrium points can be generated. The switching behavior of a hidden four-wing attractor is studied by phase portraits and time series. The dynamical properties of the multiwing attractor are analyzed via the Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. Furthermore, the hardware experiment of the proposed four-wing hidden attractors was carried out.FindingsNot only unstable equilibrium points can produce multiwing attractors but stable node–foci equilibrium points can also produce multiwing attractors. And this system can obtain 2N + 2-wing attractors as the stage pulse of the multilevel pulse function is N. Moreover, the hardware experiment matches the simulation results well.Originality/valueThis paper constructs a new multiwing chaotic system by enlarging the number of stable node–foci equilibrium points. In addition, it is a nonautonomous system that is more suitable for practical projects. And the hardware experiment is also given in this article which has not been seen before. So, this paper promotes the development of hidden multiwing chaotic attractors in nonautonomous systems and makes sense for applications.

Design of multi-wing chaotic systems with higher largest Lyapunov exponent

A multi-wing chaotic attractor with higher value of the largest Lyapunov exponent is more useful for its practical applications. This paper proposes a new design technique to generate multi-wing chaotic attractors from two-wing chaotic attractors, available in the literature. The Chen and the Lu systems are considered for demonstration. A nonlinear term of the original system is multiplied by a nonlinear function to generate multi-wings attractors. The number of wings is changed by varying the number of equilibrium points, and the equilibrium points are changed by varying the parameters of the newly added nonlinear function. The new multi-wing chaotic systems have a higher value of the largest Lyapunov exponent than their respective original systems. An interesting behavior is observed in the proposed system, i.e., the largest Lyapunov exponent increases with the variation of a system parameter. Further, the largest Lyapunov exponents of the new systems are much higher than some similar available papers.

Generating multi-wing hidden attractors with only stable node-foci via non-autonomous approach

Hidden attractor is associated with the multistability phenomenon, which has important application prospects in security communication. Recently, the study of multi-wing hidden chaotic attractor with only stable equilibria is a particular interest. By modifying simple 3D quadratic systems, this paper first proposes two systems with two symmetric equilibria. In particular, in a wide parameter region, the two systems have stable node-foci but can generate structural stable double-wing chaotic attractors, which belongs to the category of hidden attractor. The rotation matrix is applied to implement the attractor rotation. In combination with multi-level pulse sequences, multi-wing chaotic attractors with switchable stable node-foci are generated. The multi-wing hidden attractors are experimentally validated by microcontroller-based hardware experiments.

Dynamic Behavior Analysis and Robust Synchronization of a Novel Fractional-Order Chaotic System with Multiwing Attractors

To enrich the types of multiwing chaotic attractors in fractional-order chaotic systems (FOCSs), a new type of 3-dimensional FOCSs is designed in this study. The most important contribution of this FOCS consists in the coexistence of multiple multiwing chaotic attractors, including 2-wing, 3-wing, and 4-wing attractors. It is also indicated that the minimum order that the system can exhibit chaotic behavior is 0.84. Then, based on certain fractional stability criteria, a robust synchronization controller is derived for this kind of FOCSs with multiwing chaotic attractors and parametric uncertainties, and the stability of the synchronization error is proven strictly. Meanwhile, the theoretical analysis is tested by simulation results.

Design of multi-wing 3D chaotic systems with only stable equilibria or no equilibrium point using rotation symmetry

Some dissipative dynamical systems only have stable equilibria or have no equilibrium point but are able to generate chaos of non-Sil’nikov type. This category of chaotic attractors belong to hidden attractor, which is difficult to be located in the phase space. In this work, we aim at designing multi-wing 3D chaotic systems with hidden attractors. We first proposes a quadratic system with only two stable node-foci, which can generate double-wing chaotic hidden attractor. Using rotation symmetry, we construct multi-fold cover of the quadratic system and obtain multiple symmetric stable equilibria. Two examples of 2-fold and 3-fold covers of the chaotic system are presented. Furthermore, the rotation symmetry is applied to no-equilibrium system to generate multi-wing chaotic hidden attractors. By means of bifurcation diagram, finite-time Lyapunov exponents, phase portraits, and attraction basins, it is shown that the multi-wing chaotic attractors always coexist with stable point attractors. The local initial condition areas and complex boundaries for different coexisting attractors are addressed. Finally, the designed systems are implemented by analog circuit and microcontroller to verify the multi-wing hidden attractors.

Design and FPGA implementation of multi-wing chaotic switched systems based on a quadratic transformation* *Project supported by the National Natural Science Foundation of China (Grant Nos. 61973199 and 61973200) and the Taishan Scholar Project of Shandong Province of China.

Based on a quadratic transformation and a switching function, a novel multi-wing chaotic switched system is proposed. First, a 4-wing chaotic system is constructed from a 2-wing chaotic system on the basis of a quadratic transformation. Then, a switching function is designed and by adjusting the switching function, the number and the distribution of the saddle-focus equilibrium points of the switched system can be regulated. Thus, a set of chaotic switched systems, which can produce 6-to-8-12-16-wing attractors, are generated. The Lyapunov exponent spectra, bifurcation diagrams, and Poincaré maps are given to verify the existence of the chaotic attractors. Besides, the digital circuit of the multi-wing chaotic switched system is designed by using the Verilog HDL fixed-point algorithm and the state machine control. Finally, the multi-wing chaotic attractors are demonstrated via FPGA platform. The experimental results show that the number of the wings of the chaotic attractors can be expanded more effectively with the combination of the quadratic transformation and the switching function methods.

Generation of grid multi-wing chaotic attractors and its application in video secure communication system

In order to study the application of chaos in video image encryption, a real-time video secure communication system based on a new grid multi-wing chaotic system is proposed in this paper. First, by introducing sawtooth wave functions to the Lorenz system, a new grid multi-wing butterfly chaotic system with complicated dynamical behaviors is obtained. Compared with the existing multi-scroll and multi-wing chaotic systems, The system structure is simple and more easier to be implemented in a digital system. Then, a chaos-based pseudorandom random number generator is developed by implementing post-processing procedure. The present video secure communication system is designed with the closed-loop feedback scheme. The corresponding hardware implementation is developed by FPGA platforms, and the experimental results are given to verify its feasibility. Furthermore, a series of widely used secure analyses are applied to prove that the system has good security performance.

Analysis and Circuit Implementation of Fractional Order Multi-wing Hidden Attractors

Abstract In order to improve the complexity of chaotic signals, a new three-dimensional quadratic two-wing hidden chaotic system is proposed in this paper, the corresponding fractional order multi-wing system is put forward. Fractional order multi-wing chaotic hidden attractors are generated in the system. The Adomian decomposition algorithm is used to solve the proposed fractional-order chaotic system, the complex chaotic map, Lyaponov exponent spectrum, bifurcation diagram, Poincare section, power spectrum, power spectrum Shannon entropy and exponential entropy, SE (spectral entropy) complexity are obtained, and the dynamic characteristics of the system are analyzed. Also, the riddled basins of attraction and coexisting multi-stable patterns are studied. By designing the equivalent circuit module of fractional integral operator, a circuit system for the fractional-order chaotic system is constructed and the 0.9-order multi-wing chaotic attractors are realized.

A Five-Dimensional Grid Multi-wing Butterfly Chaotic System and Its Circuit Simulation

Based on the two-wing butterfly chaotic system, through adding two first-order differential equations with the piecewise linear function, a five-dimensional grid multi-wing butterfly chaotic system is presented. Through numerical simulation, it is studied for the gird multi-wing butterfly chaotic attractors, the Lyapunov exponent spectrum and the bifurcation diagram of the system. At the end, an electronic circuit is designed to carry out the system. The circuit experimental results are consistent with the results of numerical simulation, which confirm the availability and feasibility of this approach.

Creation and circuit implementation of two‐to‐eight–wing chaotic attractors using a 3D memristor‐based system

SummaryThis paper constructs a three‐dimensional (3D) memristor‐based system for creating multiwing chaotic attractors. A second‐degree polynomial memristance function and a sixth‐order exponent internal state memristor function with one parameter are employed, and the complexity of attractors is increased. A detailed analysis on dynamical behaviors of the proposed system are described, such as the bifurcation diagrams, finite‐time local Lyapunov exponents, time series, phase portraits, and Poincaré maps. By adjusting the design parameters, the system displays two‐to‐eight–wing chaotic attractors, especially the five‐wing and seven‐wing attractors, which have never been found in the known systems. Further, we provide the calculation formula of the number of wings in the system, discuss the distribution of the involving inner holes on the plane, and design an electronic circuit to realize the proposed system. The experimental results of the circuit implementation agree with the numerical simulations on Matlab well. It indicates the potential engineering applications for various chaos‐based information systems.

3D Grid Multi-Wing Chaotic Attractors

As reported in the existing literature, wing attractors are confined to 1D [Formula: see text]-wing attractors, 2D [Formula: see text]-grid wing attractors. In this paper, we break this limitation and generate 3D [Formula: see text]-grid multi-wing chaotic attractors (GMWCAs). The 3D GMWCAs are produced via the following three steps: (1) applying rotation transformation to a double-wing Lorenz-like system to ensure that its saddle-focus equilibria with index 2 are located on the plane [Formula: see text]; (2) extending the wing attractors of the transformed Lorenz-like system along the [Formula: see text]-axis to have mirror symmetry; (3) introducing stair switching functions to increase the number of saddle-focus equilibria with index 2 along the [Formula: see text]-axis and [Formula: see text]-axis. Furthermore, some basic dynamical properties of the 3D chaotic system, including equilibria, symmetry, dissipativity, Lyapunov exponents and bifurcation diagram, are investigated and a module-based unified circuit diagram is designed. The effectiveness of this approach is confirmed by both numerical simulations and electrical circuit experiment.

Design and control of a multi-wing dissipative chaotic system

Considering the dissipative condition, a multi-wing chaotic attractor with a unique perspective is designed, using geometry root locus. By using a nonlinear dissipative term, a new structure is proposed to obtain a dissipative chaotic system with a four-wing attractor. The system properties, namely the phase portraits, bifurcation diagrams, Lyapunov exponents spectrum, Poincare maps and local stability, are investigated by numerical simulations. The phenomenon of multiple attractors is discussed and two double-wing smooth chaotic attractors using two different initial conditions are generated. The nonlinear time-delayed inputs are proposed for chaos control of the new system via bifurcation theory. Based on center manifold and normal form theories, the procedure of determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated.

A multi-wing spherical chaotic system using fractal process

In this paper, a multi-wing spherical chaotic system is derived via a fractal process based on Qi 3D four-wing chaotic system. The system can generate a 4n-wing chaotic system. Numerical simulations demonstrate the validity and feasibility of the proposed method, which may generate multi-wing chaotic systems not only using the Qi 3D four-wing system but also other 3D autonomous chaotic systems. Compared with other multi-wing chaotic attractors, the proposed multi-wing chaotic attractors are much easier to adjust the number of the wings. Hamiltonian energy formulas of both original system and the transformed system are obtained, which concludes that the energy is decreased as the multi-wing number increased. Poincare map and bifurcation analysis show that the newly generated system has extremely rich dynamics and the topological structure is much more complicated than the original system. The 4n-wing chaotic system is more suitable for the further research on the application of chaos encryption than the original chaotic system.