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.     

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.     

Bifurcations in a piecewise linear system

We study two bifurcations which, because of the piecewise linear nature of the system under consideration, occur at the same parameter value. The three orbits created in this compound bifurcation are the principal periodic orbits of a homoclinic bifurcation seen in the system.     

Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.     

A new chaotic Hopfield network with piecewise linear activation function

This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectrum. Numerical simulations show that the network displays chaotic behaviours for some well selected parameters.     

Invariant curves,attractors, and phase diagram of a piecewise linear map with chaos

We introduce equations describing the invariant curves associated with periodic points in a wide class of two-dimensional invertible maps, which in the special case of the mapT(x, z)=(1?a¦x¦+bz,x) can be solved by analytical methods. In the dissipative case several branches of the separatrices of the fixed points, as well as, of the period-2 and -4 points, are constructed. The regions of the parameter space where a given type of strange attractor exists are located. We point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period-2 points and the stable manifold of the fixed point may occur separately, and the latter leads already to the appearance of a two-piece strange attractor. This phenomenon may happen at weak dissipation in other maps, too. In the conservative caseb=1 separatrices and certain invariant tori are calculated.     

Interpretation of experimental data based on the piecewise linear approximation measurement model

Methods for estimating the input signal of a measuring device are considered. Information about the device is contained in the measurement results of a set of known test signals and the measurements were obtained in natural or in computational experiments. A method is proposed for the approximation of the measurement model by means of a piecewise linear model that is consistent with the results of measurements and the method of evaluation of the input signal and its accuracy. An example is given of the solution of a problem of the interpretation of measurements based on the approximation of a model of the photosynthetic system, in which the measured parameters are the values of saturation ΔpH and the rate of ATP synthesis and the estimated parameters are the concentration of photosystem-2 and light intensity.     

A new 5D hyperchaotic system with stable equilibrium point,transient chaotic behaviour and its fractional-order form

In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and five equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numerically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coexisting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system.     

基于线性控制的分数阶统一混沌系统的同步

研究了分数阶统一混沌系统的同步.基于分数阶稳定性原理,提出了通过线性反馈实现分数阶统一混沌系统的同步方法.所设计的控制器为单一控制变量的线性控制器,且不需要计算反馈系数.通过对分数阶Lorenz混沌系统、Chen混沌系统和Lü混沌系统的数值模拟,证实了所提方法的有效性.     

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.     

分数阶共轭Chen混沌系统中的混沌及其电路实验仿真

研究了分数阶共轭Chen混沌系统的混沌行为, 给出了分数阶共轭Chen系统存在混沌吸引子的必要条件. 并设计了一种传递函数的新电路单元用来实现分数阶共轭Chen混沌系统, 利用Multisim2001电子工作平台进行了电路实验仿真. 数值模拟和电路实验仿真结果相符, 证实了分数阶共轭Chen系统确实存在混沌现象和所设计电路单元的有效性. 关键词： 分数阶 共轭Chen混沌系统 电路单元 电路仿真     

A periodically forced piecewise linear oscillator

A single-degree of freedom non-linear oscillator is considered. The non-linearity is in the restoring force and is piecewise linear with a single change in slope. Such oscillators provide models for mechanical systems in which components make intermittent contact. A limiting case in which one slope approaches infinity, an impact oscillator, is also considered. Harmonic, subharmonic, and chaotic motions are found to exist and the bifurcations leading to them are analyzed.     

一个新分数阶超混沌系统及其混沌同步

给出了一个新的分数阶超混沌系统,利用严格数学理论实现了该分数阶超混沌系统的混沌同步,在同步过程中并未删除响应系统的非线性项,理论分析与仿真计算表明了同步方法的有效性. 关键词： 分数阶超混沌系统 混沌同步 非线性项     

A new image encryption algorithm based on fractional-order hyperchaotic Lorenz system

We propose a new image encryption algorithm on a basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded in the proposed algorithm to enhance the security. Such an algorithm is detailed in terms of security analyses, including correlation analysis, information entropy analysis, run statistic analysis, mean-variance gray value analysis, and key sensitivity analysis. The experimental results demonstrate that the proposed image encryption scheme has the advantages of large key space and high security for practical image encryption.     

一种新的分段混沌系统的同步控制

利用状态观测器思想构造驱动和响应系统，实现了一个新近提出的分段线性混沌系统的同步控制，并且推导了两个不同初值的系统实现同步的充分条件.理论分析说明该方法对原系统结构没有特殊要求，无需求解Lyapunov函数，适合应用于分段混沌系统.仿真实验的良好结果验证了该算法的有效性，对于不同初值的分段系统均能实现快速同步.同时该算法适用于一类分段混沌系统的同步控制. 关键词： 分段线性混沌系统 状态观测器 同步控制     

Circuit implementation of a new hyperchaos in fractional-order system

This paper introduces a new four-dimensional （4D） hyperchaotic system, which has only two quadratic nonlinearity parameters but with a complex topological structure. Some complicated dynamical properties are then investigated in detail by using bifurcations, Poincare mapping, LE spectra. Furthermore, a simple fourth-order electronic circuit is designed for hardware implementation of the 4D hyperchaotic attractors. In particular, a remarkable fractional-order circuit diagram is designed for physically verifying the hyperchaotic attractors existing not only in the integer-order system but also in the fractional-order system with an order as low as 3.6.     

A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system

We propose a new image encryption algorithm on the basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded in the proposed algorithm to enhance the security. Such an algorithm is detailed in terms of security analyses, including correlation analysis, information entropy analysis, run statistic analysis, mean-variance gray value analysis, and key sensitivity analysis. The experimental results demonstrate that the proposed image encryption scheme has the advantages of large key space and high security for practical image encryption.     

一个新的分数阶混沌系统的翼倍增及滑模同步

首先,提出了一个新的分数阶混沌系统,通过对系统第二个等式的线性项x作绝对值运算,并分析了其唯一的参数k,该参数在一定区间内取值时可将混沌吸引子由两个翼的结构变换为四翼的拓扑结构,从而实现翼倍增. 其次,分别采用Matlab和Multisim对新的分数阶系统及其翼倍增系统进行了数值模拟和电路仿真,电路仿真结果和数值模拟结果相一致. 最后,基于滑模变结构控制理论和分数阶稳定性定理,为新的分数阶系统及其翼倍增系统设计了新的分数阶积分滑模控制器实现系统的同步,仿真结果和理论分析相一致,证实了所设计滑模控制器的有效性. 关键词： 分数阶 翼倍增 滑模控制 同步     

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.     

On global attractors for a class of nonhyperbolic piecewise affine maps

Dynamical behavior of a class of nonhyperbolic discrete systems are considered. These systems are generated by iterating planar maps that are piecewise isometries, and they arise as mathematical models for signal processing, digital filters and modulator dynamics. Planar piecewise isometries may be discontinuous and/or non-invertible. First, the authors consider attraction caused by discontinuity in planar piecewise isometries. Namely, they have shown that the maximal invariant set can induce an invariant measure, and all the Lyapunov exponents are zero under this invariant measure. Second, they discuss various definitions of global attractors and their existence and uniqueness for discontinuous maps, and introduce a few examples in which the attractors are created due to discontinuity. Third, they study the relation between invariance and invertibility for various nonhyperbolic maps, and finally they investigate decomposability of global attractors for certain nonhyperbolic systems.