Design and analysis of multiscroll chaotic attractors from saturated function series

This paper initiates a saturated function series approach for chaos generation. The systematic saturated function series methodology developed here can create multiscroll chaotic attractors from a three-dimensional (3D) linear autonomous system with a simple saturated function series controller, including one-directional n-scroll, two-directional n/spl times/m-grid scroll, and 3-D n/spl times/m/spl times/l-grid scroll chaotic attractors. The dynamical behaviors and chaos generation mechanism of multiscroll systems are further investigated by analyzing the system trajectories. In particular, a two-dimensional (2D) Poincare/spl acute/ return map is rigorously derived for verifying the chaotic behaviors of the double-scroll chaotic attractor, which is a basic generator of various multiscroll chaotic attractors investigated in the paper.     

Design and implementation of n-scroll chaotic attractors from a general jerk circuit

This paper proposes a novel nonlinear modulating function approach for generating n-scroll chaotic attractors based on a general jerk circuit. The systematic nonlinear modulating function methodology developed here can arbitrarily design the swings, widths, slopes, breakpoints, equilibrium points, shapes, and even the general phase portraits of the n-scroll chaotic attractors by using the adjustable sawtooth wave, triangular wave, and transconductor wave functions. The dynamic mechanism and chaos generation condition of the general jerk circuit are further investigated by analyzing the system stability. A simple block circuit diagram, including integrator, sawtooth wave and triangular wave generators, buffer, switch linkages, and voltage-current conversion resistors, is designed for the hardware implementations of various 3-12-scroll chaotic attractors via switchings of the switch linkages. This is the first time to experimentally verify a 12-scroll chaotic attractor generated by an analog circuit. In particular, the recursive formulas of system parameters and real physical circuit parameters are rigorously derived for the hardware implementations of the n-scroll chaotic attractors. Moreover, the adjustability of the nonlinear modulating function and the rigorous recursive formulas together provide a theoretical principle for the hardware implementations of various chaotic attractors with a large number of scrolls.     

A new multi-scroll Chua’s circuit with composite hyperbolic tangent-cubic nonlinearity: Complex dynamics,Hardware implementation and Image encryption application

In this paper, a new method for generating multi-scroll chaotic attractors via constructing a compound hyperbolic tangent-cubic nonlinear function in canonical Chua’s circuit is presented. The basic dynamic characteristics of the system are analyzed, including equilibrium points, bifurcation diagrams, Lyapunov exponents, phase portraits, time-domain diagrams and attractive basins. What is surprising is that the proposed multi-scroll Chua’s circuit also exhibits rich dynamic behaviors like coexisting multiple attractors, transient period, intermittent chaos and offset boosting. In addition, we put forward the application of the system in chaotic image encryption, and analyzed some security performance evaluation indexes to show that the new Chua’s chaotic cipher system has high security and reliable encryption performance. Finally, the hardware design and experiments of the chaotic digital circuits and image encryption are carried out. Both numerical simulation and FPGA experimental results verify the feasibility and usability of the proposed new multi-scroll Chua’s system.     

n-scroll chaotic oscillators by second-order systems and double-hysteresis blocks

A new circuit design for generating n-scroll chaotic oscillators using a linear second-order unstable system with double-hysteresis series is proposed. The hysteresis series is realised by building blocks. With this proposed scheme, the n-scroll chaotic attractors can be easily generated along the directions of the system state variables     

四维系统中多涡卷混沌吸引子及其同步的研究

通过四维系统中的多涡卷混沌吸引子的研究，提出了一个简便数学公式来确定产生多涡卷的分段线性伏安特性的转折点电压，该方法可产生任意个数的涡卷。介绍了以此方法确定了产生6-8涡卷的混沌振荡电路，并对6-8涡卷的混沌振荡电路进行了仿真。最后用脉冲同步法对5涡卷混沌振荡电路进行了同步仿真研究。     

忆阻开关混沌电路及其吸引子共存现象研究

为了研究忆阻开关电路的动力学行为,该文提出一种具有多吸引子共存现象的忆阻开关混沌电路。在该电路中存在多吸引子分岔,当系统中发生边界碰撞之后,系统中将产生不同的吸引子共存现象。其中包括单周期极限环与混沌吸引子共存,不同的混沌吸引子共存,对称的2周期极限环共存现象,以及对称的2周期极限环与5周期极限环共存现象等。该文通过相图、分岔图等数值仿真,分析了该电路的动力学行为,并利用PSIM电路仿真验证了其电路的可行性,对开关电路中多吸引子共存现象和混沌应用的研究具有重要意义。     

Comparison of the effects of nonlinear gain and weak opticalfeedback on the dynamics of semiconductor lasers

The influence of nonlinear gain and optical feedback on the dynamics of single-mode semiconductor lasers are numerically investigated based on the Lang and Kobayashi model. It is well known that the nonlinear gain tends to stabilize the dynamics, while the optical feedback tends to increase the instabilities. In this paper, we study the behavior of the attractors when the feedback level k and the gain saturation coefficient ε vary and show that the effects of these parameters are surprisingly opposite. For example, we find that the route to chaos that the external cavity modes follow for increasing k is reversed for increasing ε in an almost identical manner. When the feedback increases the modes follow the usual quasi-periodic route and turn into torus. If k continues to increase, the torus become chaotic attractors as the result of several period-doubling bifurcations or a third Hopf bifurcation. Further increase of k causes the chaotic attractors to lose stability, Contrarily, if the value of the parameter ε is increased, the attractors recover their stability and reverse the route becoming simple torus again. If ε is increased further, the torus reverse the quasi-periodic route and turn into stable modes again. We also find that on the contrary to k, the parameter ε enhances the stability region of an attractor. We show that the feedback level above which a limit cycle emerges from a stable mode, the feedback level above which a torus emerges from a limit cycle, and the feedback level above which a chaotic attractor loses stability are all increasing functions of ε     

一个四维四翼混沌系统及其电路实现

一个四翼混沌吸引子通过引入了两个线性反馈控制量构成一个新的四维四翼混沌系统,吸引子更加稳定、可靠,易于电路实现。文章分析了该四维四翼混沌系统的基本特性,并为实现该新系统设计了一个模拟电路,实验结果表明:数值仿真和电路实现具有很好的一致性。该系统有利于保密通信等基于混沌的实际应用。     

基于二极管的限幅蔡氏电路实验研究

根据经典蔡氏电路的数学模型,获得系统各个变量之间的运算关系,利用运算放大器、电阻、电容及二极管等元器件,完成非线性限幅函数的电路设计,实现无电感元件的等效蔡氏电路硬件实验图。结果表明,通过调试电路中电阻值,可以直接从示波器观测到蔡氏电路的三个运动变量,包括周期、单涡卷、双涡卷等运动行为的时序图和相位图。搭建的电路结构简单,学生实验时调试容易,并且由于电路中不含电感元件,实验过程中混沌吸引子的稳定性也明显得到了提高。     

Novel unified chaotic system with multi-parameter invariable Lyapunov exponent spectrum

Based on the traditional Qi chaotic system,a novel unified chaotic system with the complex chaotic characteristics was constructed by adding the control parameters and modifying the nonlinear terms.Firstly,basic dynamical characteristics of chaotic system were analyzed,and phase portrait,time domain waveform diagram,Poincare mapping and power spectrum diagram were numerically simulated.Secondly,system parameters influence on chaotic system was discussed through Lyapunov exponent spectrum,bifurcation diagrams and chaotic signal amplitude.It was found that the unified chaotic system can generate the four new types of two-wing chaotic attractors with the multi-parameter invariable Lyapunov exponent spectrum characteristics.Meanwhile,there exist the functions of the global and local nonlinear amplitude modulation parameters.Thirdly,taking the first chaotic attractor of system as an example by introducing the two new types of nonlinear functions,the expansion of grid multi-wing attractor was realized.Finally,the hardware circuit of novel unified chaotic system was constructed.The four new types of chaotic attractors are observed experimentally,which is consistent with numerical simulation results and verified the feasibility of the proposed system.     

n×m涡卷混沌吸引子的研究及硬件实现

提出了产生n×m涡卷混沌吸引子的一种新方法,用三角波函数序列和阶跃函数序列构造网格多涡卷混沌系统,找出函数中平衡点和转折点的值,并对系统的基本动力学特性进行了研究,包括理论分析和计算机模拟仿真。用模块化的方法设计了现场可编程门阵列（FPGA）硬件实验电路,给出了相关的实验结果。计算机仿真和FPGA硬件实验结果相吻合,由此证明了该系统的混沌特性。     

Hyperchaotic spiking oscillators with periodic pulse-train input

This paper studies simple spiking oscillators with periodic pulse-train input. The circuits are piecewise linear and a normal form circuit equation is derived in order to extract essential parameters. The return map and its Jacobian matrix can be described precisely. If the input is not present, the circuits can exhibit hyperchaotic and periodic attractors. As the input is applied, the attractors can be changed into various periodic, chaotic and hyperchaotic attractors. The phenomena are characterized by Lyapunov exponents of the return map. Typical phenomena can be confirmed experimentally. These results provide basic information for analysis of bifurcation phenomena and application to pulse-coupled networks.     

多涡卷Jerk混沌吸引子的EWB仿真

混沌吸引子的产生是将混沌信号应用于加密领域的基础。文中在双涡卷Jerk混沌吸引子的基础上,利用阶梯波函数构造产生出多涡卷Jerk混沌吸引子,分析了多涡卷Jerk混沌吸引子的产生机理,并根据无量纲状态方程,分别设计了6-涡卷和7-涡卷混沌吸引子仿真电路。利用EWB电路仿真软件进行了仿真验证,仿真结果与数学分析结果一致,证明了多涡卷Jerk系统的存在。     

A new hidden chaotic attractor with extreme multi-stability

The topic of hidden attractors is a very hot topic in nonlinear dynamics. Multi-stability is another important topic in that area. One recent important category of multi-stable systems is “systems with extreme multi-stability”. In this paper we introduce a new five-dimensional chaotic system with both hidden attractors and extreme multi-stability. Such systems are very rare in literature. Also, we show its feasibility by the help of electronic circuit implementation.     

一个混沌系统性质分析及其FPGA电路实现

构造了一个新的三维连续自治混沌系统,该系统含有3个参数,2个非线性乘积项通过理论推导、数值仿真、Lyapunov指数谱、分岔图研究了系统的基本动力学特性,还利用FPGA技术设计了一个数字电路,实现三维混沌系统,实验结果与仿真结果完全一致,提出了一种基于FPGA技术实现混沌系统的新方法     

一种四螺旋混沌系统仿真研究

为了改善Lü系统的混沌特性,通过在混沌Lü系统上添加一维线性状态反馈控制器,构建了一个新的四维连续自治混沌系统,得到了奇异的四螺旋鲁棒混沌吸引子。利用相轨图、Lyapunov指数谱等数值方法,其最大Lyapunov指数为正值且几乎恒定不变,系统具有很好的鲁棒性.随控制参数变化,系统能够从四螺旋混沌吸引子演变成双螺旋混沌吸引子。通过在混沌Lü系统电路上外加一个简单的线性积分电路实现了新混沌系统的实验电路,实验仿真与数值方法所得结果有着很好的一致性。     

On Multiscroll Chaotic Attractors in Hysteresis-Based Piecewise-Linear Systems

This brief analyzes the multi scroll chaotic attractors in hysteresis-based piecewise-linear systems. The dynamics of the chaotic attractors are first analyzed. The unstable limit cycles and the boundary conditions for the occurrences of chaotic behaviors are derived. The sensitivity of systems parameters on the chaotic behaviors is also studied, accompanied by a number of simulations.     

一种新型双忆阻混沌系统动力学及其电路实现研究

在经典的蔡氏混沌电路基础上,引入三次非线性磁控忆阻模型,利用一个磁控忆阻模型和一个荷控忆阻模型,外加一个负电导替换变形蔡氏电路中的蔡氏二极管,设计了一个五阶混沌电路,用常规的方法研究系统的基本动力学特性。通过数值仿真结果表明电路在参数变化情况下能产生Hopf分岔和反倍周期分岔两种分岔行为,并能产生双涡卷、单涡卷、极限环、同宿轨等不同轨道,出现了双单摆运动。观察混沌吸引子推广到功率与能量信号,观察到蝴蝶翅膀重叠的奇异吸引子。通过改变初始值,能产生共存吸引子和周期极限环共存现象。为了验证电路的混沌行为,将对设计的电路进行了PSpice仿真,电路仿真结果验证了理论分析的正确性。     

A wafer-scale 3-D circuit integration technology

The rationale and development of a wafer-scale three-dimensional (3-D) integrated circuit technology are described. The essential elements of the 3-D technology are integrated circuit fabrication on silicon-on-insulator wafers, precision wafer-wafer alignment using an in-house-developed alignment system, low-temperature wafer-wafer bonding to transfer and stack active circuit layers, and interconnection of the circuit layers with dense-vertical connections with sub-/spl Omega/ 3-D via resistances. The 3-D integration process is described as well as the properties of the four enabling technologies. The wafer-scale 3-D technology imposes constraints on the placement of the first lithographic level in a wafer-stepper process. Control of wafer distortion and wafer bow is required to achieve submicrometer vertical vias. Three-tier digital and analog 3-D circuits were designed and fabricated. The performance characteristics of a 3-D ring oscillator, a 1024 /spl times/ 1024 visible imager with an 8-/spl mu/m pixel pitch, and a 64 /spl times/ 64 Geiger-mode laser radar chip are described.     

Multi-scroll hyperchaotic system based on colpitts model and its circuit implementation

Based on the framework of Colpitts oscillator, a four-dimensional multi-scroll hyperchaotic system is proposed, which generates (2M+1)×(2N+1)-scroll chaotic and hyperchaotic attractors. The key strategy is to increase the number of index-2 equilibrium points by introducing two unit saw-tooth functions to extend and modify the Colpitts oscillator model. By using bifurcation diagram and phase portrait, the dynamical characteristics of the multi-scroll hyperchaotic system are briefly studied. Moreover, micro-controller based circuit realization is introduced and the experimental results demonstrate that 7×5-scroll chaotic and hyperchaotic attractors can be obtained in the digital circuit.