Design and FPGA implementation of a memristor-based multi-scroll hyperchaotic system

A novel memristor-based multi-scroll hyperchaotic system is proposed. Based on a voltage-controlled memristor and a modulating sine nonlinear function, a novel method is proposed to generate the multi-scroll hyperchaotic attractors. Firstly, a multi-scroll chaotic system is constructed from a three-dimensional chaotic system by designing a modulating sine nonlinear function. Then, a voltage-controlled memristor is introduced into the above-designed multi-scroll chaotic system. Thus, a memristor-based multi-scroll hyperchaotic system is generated, and this hyperchaotic system can produce various coexisting hyperchaotic attractors with different topological structures. Moreover, different number of scrolls and different topological attractors can be obtained by varying the initial conditions of this system without changing the system parameters. The Lyapunov exponents, bifurcation diagrams and basins of attraction are given to analyze the dynamical characteristics of the multi-scroll hyperchaotic system. Besides, the field programmable gate array (FPGA) based digital implementation of the memristor-based multi-scroll hyperchaotic system is carried out. The experimental results of the FPGA-based digital circuit are displayed on the oscilloscope.     

不同类型混沌吸引子的复合

为了实现不同类型混沌吸引子之间的复合,采用理论分析、数值仿真和电路仿真方法,通过设计合适的切换控制器实现了不同两涡卷混沌系统之间的复合、不同多涡卷混沌系统之间的复合、两涡卷混沌系统与两翅膀混沌系统之间的复合和多涡卷混沌系统与多翅膀混沌系统之间的复合.通过观察吸引子相图、最大Lyapunov指数和Poincaré截面,分析了复合系统的动力学行为.设计了复合多涡卷-多翅膀吸引子的模拟电路,并对其进行了电路仿真,得到的电路仿真结果与数值仿真结果相一致.这表明利用切换控制器实现不同类型混沌系统之间复合方法的正确性.     

基于同相第二代电流传输器的网格多涡卷混沌电路研究

通过简化广义Jerk系统,构建了一个三维网格多涡卷混沌系统,利用同相第二代电流传输器(CCII+)设计并实现了该系统的模拟电路,将该电路与传统运放实现的混沌电路进行了对比,发现基于CCII+的混沌电路能在更高频段产生网格多涡卷吸引子,且电路结构更简单,使用器件更少,最后,通过示波器观测到了网格多涡卷混沌吸引子,验证了混沌电路的物理可实现性,电路实验结果与数值仿真结果一致。     

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 Poincaré 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 Poincaré 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.     

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincar'e map of the system is constructed. Using the Poincar'e map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.     

Impact of autocatalysis chemical reaction on nonlinear radiative heat transfer of unsteady three-dimensional Eyring–Powell magneto-nanofluid flow

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.     

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.     

Dynamical behaviors of a chaotic system with no equilibria

Based on Sprott D system, a simple three-dimensional autonomous system with no equilibria is reported. The remarkable particularity of the system is that there exists a constant controller, which can adjust the type of chaotic attractors. It is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping and period-doubling route to chaos are analyzed with careful numerical simulations.     

Analysis and implementation of new fractional-order multi-scroll hidden attractors

To improve the complexity of chaotic signals,in this paper we first put forward a new three-dimensional quadratic fractional-order multi-scroll hidden chaotic system,then we use the Adomian decomposition algorithm to solve the proposed fractional-order chaotic system and obtain the chaotic phase diagrams of different orders,as well as the Lyaponov exponent spectrum,bifurcation diagram,and SE complexity of the 0.99-order system.In the process of analyzing the system,we find that the system possesses the dynamic behaviors of hidden attractors and hidden bifurcations.Next,we also propose a method of using the Lyapunov exponents to describe the basins of attraction of the chaotic system in the matlab environment for the first time,and obtain the basins of attraction under different order conditions.Finally,we construct an analog circuit system of the fractional-order chaotic system by using an equivalent circuit module of the fractional-order integral operators,thus realizing the 0.9-order multi-scroll hidden chaotic attractors.     

A novel multi-directional multi-scroll chaotic system and its CCII+ circuit implementation

This paper proposes a generalized formula of multi-directional multi-scroll chaotic system. Based on the generalized formula, by adding the nonlinear functions, and controlling the parameters of system, multi-directional multi-scroll chaotic attractors with different order system can be generated. According to this method, the correctness of generalized formula is verified. Then, basic dynamic characteristics of the system including the fractal, equilibrium points, Poincaré section, Lyapunov exponent spectrum and bifurcation diagram are analyzed. Finally, a nine-dimensional chaotic circuit based on CCII₊ devices is designed, which verified the feasibility of the generalized formula and multi-directional multi-scroll chaotic circuit with CCII₊ devices.     

Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system

A novel 5-dimensional(5D) memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing 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 Poincar′e 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.     

Adaptive generalized synchronization between Chen system and a multi-scroll chaotic system

Based on Lyapunov theory, the adaptive generalized synchronization between Chen system and a multi-scroll chaotic system is investigated. According to the form of target function a proper adaptive controller is designed, by which the controlled Chen system can be synchronized with a multi-scroll chaotic system including unknown parameters. The Lyapunov direct method is exploited to prove that the synchronization error and parameter identification error both converge to zero. Numerical simulation results verify the feasibility of the proposed method further.     

The analysis of a novel 3-D autonomous system and circuit implementation

This Letter presents a new three-dimensional autonomous system with four quadratic terms. The system with five equilibrium points has complex chaotic dynamics behaviors. It can generate many different single chaotic attractors and double coexisting chaotic attractors over a large range of parameters. We observe that these chaotic attractors were rarely reported in previous work. The complex dynamical behaviors of the system are further investigated by means of phase portraits, Lyapunov exponents spectrum, Lyapunov dimension, dissipativeness of system, bifurcation diagram and Poincaré map. The physical circuit experimental results of the chaotic attractors show agreement with numerical simulations. More importantly, the analysis of frequency spectrum shows that the novel system has a broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

分数阶Lorenz系统的分析及电路实现

频域传递函数近似方法不仅是常用的 分数阶混沌系统相轨迹的数值分析方法之一, 而且也是设计分数阶混沌系统电路的主要方法. 应用该方法首先研究了分数阶Lorenz系统的混沌特性, 通过对Lyapunov指数图、分岔图和数值仿真分析, 发现了其较为丰富的动态特性, 即当分数阶次从0.7到0.9以步长0.1变化时, 该分数阶Lorenz系统既存在混沌特性, 又存在周期特性, 从数值分析上说明了在更低维的Lorenz系统中存在着混沌现象. 然后又基于该方法和整数阶混沌电路的设计方法, 设计了一个模拟电路实现了该分数阶Lorenz系统, 电路中的电阻和电容等数值是由系统参数和频域传递函数近似确定的. 通过示波器观测到了该分数阶Lorenz系统的混沌吸引子和周期吸引子的相轨迹图, 这些电路实验结果与数值仿真分析是一致的, 进一步从物理实现上说明了其混沌特性. 关键词： 分数阶系统 Lorenz系统 分岔分析 电路实现     

基于Julia分形的多涡卷忆阻混沌系统

忆阻器作为一种非线性电子元件,能用作混沌系统中的非线性项,从而提高系统的复杂度.分形与混沌是密切相连的,分别对两者的研究都已成熟,却鲜有将分形过程应用到混沌系统中,以产生丰富的混沌吸引子.为了探索将分形与混沌系统相结合的可能性,本文首先提出了一个新的忆阻混沌系统,并从对称性、耗散性、平衡点稳定性、功率谱、Lyapunov指数和分数维等方面探讨了系统的动力学特性;紧接着,把经典的Julia分形过程应用到该忆阻混沌系统中,产生了新的混沌吸引子,并将几种由Julia分形衍生的变形Julia分形过程应用于文中提出的忆阻混沌系统,获得了丰富的混沌吸引子;最后,讨论了分形过程中的复常数对系统的影响.从仿真结果可以看出,分形过程与混沌系统的结合能产生丰富的多涡卷混沌吸引子.这不仅为产生多涡卷混沌吸引子提供了一种新方法,还弥补了使用功能函数方法造成混沌系统不光滑的不足.     

Dynamics analysis and synchronization of a new chaotic attractor

In this paper, a new simple chaotic system is discussed. Basic dynamical properties of the new attractor are demonstrated in terms of phase portraits, equilibria and stability, Lyapunov exponents, a dissipative system, Poincaré mapping, bifurcation diagram, especially Hopf bifurcation. Next, based on well-known Lyapunov stability theorem, backstepping design is proposed for synchronization of the new chaotic system. At last, numerical studies are provided to illustrate the effectiveness of the presented scheme.     

The analysis of complex behaviours of a novel three dimensional autonomous system

This paper introduces a new three dimensional autonomous system with five equilibrium points.It demonstrates complex chaotic behaviours within a wide range of parameters,which are described by phase portraits,Lyapunov exponents,frequency spectrum,etc.Analysis of the bifurcation and Poincar’e map is used to reveal mechanisms of generating these complicated phenomena.The corresponding electronic circuits are designed,exhibiting experimental chaotic attractors in accord with numerical simulations.Since frequency spectrum analysis shows a broad frequency bandwidth,this system has perspective of potential applications in such engineering fields as secure communication.     

Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations

In this paper, the bifurcations and chaotic motions of higher-dimensional nonlinear systems are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam. The Kelvin viscoelastic model is chosen to describe the viscoelastic property of the beam material. Firstly, the nonlinear governing equations of nonplanar motion for an axially accelerating moving viscoelastic beam are established by using the generalized Hamilton’s principle for the first time. Then, based on the Galerkin’s discretization, the governing equations of nonplanar motion are simplified to a six-degree-of-freedom nonlinear system and a three-degree-of-freedom nonlinear system with parametric excitation, respectively. At last, numerical simulations, including the Poincare map, phase portrait and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the axially accelerating moving viscoelastic beam. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity, the amplitude of velocity fluctuation and the frequency of velocity fluctuation are respectively presented when other parameters are fixed. The Lyapunov exponents are calculated to identify the existence of the chaotic motions. From the numerical results, it is indicated that the periodic, quasi-periodic and chaotic motions occur for the nonplanar nonlinear vibrations of the axially accelerating moving viscoelastic beam. Observing the in-plane nonlinear vibrations of the axially accelerating moving viscoelastic beam from the numerical results, it is found that the nonlinear responses of the six-degree-of-freedom nonlinear system are much different from that of the three-degree-of-freedom nonlinear system when all parameters are same.     

四阶Colpitts混沌振荡器

提出了一种新的四阶Colpitts混沌振荡器.理论设计与电路实验表明,在三阶Colpitts混沌振荡器中的电感两端并联一个电容器C₃,可构建出一种四阶Colpitts混沌振荡器.当C₃的取值变化时,电路的谐振频率随之改变,从而使该振荡器经过倍周期分岔进入混沌状态.对四阶Colpitts混沌振荡器的平衡点、分岔和李氏指数等基本动力学问题进行了分析.最后通过数值仿真和电路实验证实了这一方法的可行性. 关键词： 四阶Colpitts混沌振荡器 混沌吸引子 电路实现     

Generation of grid multi-scroll chaotic attractors via switching piecewise linear controller

In this Letter, a novel method is developed for generating grid multi-scroll chaotic attractors using switching piecewise linear controller. First, a third-order linear system is designed to ensure that its unique equilibrium point belongs to a saddle-focus type with index 2 and the corresponding eigenvalues satisfy Shilnikov conditions. Then, by three different types of switching control strategies, the equilibrium point can be extended along both xy plane and z axis direction, so as to generate grid multi-scroll chaotic attractors. The dynamical behaviors are further analyzed. Moreover, an improved module-based circuit is designed for realizing 5×3 and 4×4 grid scroll chaotic attractors, and the experimental results are also obtained, which is consistent with the numerical simulations.