A Simple Parallel Chaotic Circuit Based on Memristor

This paper reports a simple parallel chaotic circuit with only four circuit elements: a capacitor, an inductor, a thermistor, and a linear negative resistor. The proposed system was analyzed with MATLAB R2018 through some numerical methods, such as largest Lyapunov exponent spectrum (LLE), phase diagram, Poincaré map, dynamic map, and time-domain waveform. The results revealed 11 kinds of chaotic attractors, 4 kinds of periodic attractors, and some attractive characteristics (such as coexistence attractors and transient transition behaviors). In addition, spectral entropy and sample entropy are adopted to analyze the phenomenon of coexisting attractors. The theoretical analysis and numerical simulation demonstrate that the system has rich dynamic characteristics.     

The electric double layer effect on the microchannel flow stability and heat transfer

The effect of the electric double layer (EDL) on the linear stability of Poiseuille planar channel flow is reported. It is shown that the EDL destabilises the linear modes, and that the critical Reynolds number decreases significantly when the thickness of the double layer becomes comparable with the height of the channel. First results coming from direct numerical simulations on the non-linear effects show also that the by-pass transition is much more rapid in the presence of EDL. There is an acceptable qualitative correspondence between the estimated transitional Reynolds numbers and some experiments, showing that early transition is plausible in microchannels under some conditions. Several questions remain however unanswered such as the surface conduction effect on EDL.     

Multistability and coexisting transient chaos in a simple memcapacitive system

The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper. The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix. Besides, complex dynamical behaviors are investigated in the system, such as coexisting attractors, hidden attractors,coexisting bifurcation modes, intermittent chaos, and multistability. From the theoretical analyses and numerical simulations, it is found that there are four different kinds of transient transition behaviors in the memcapacitive system. Finally,field programmable gate array(FPGA) is used to implement the proposed chaotic system.     

低阶格式在大涡模拟计算中的适用性

采用三维Taylor-Green涡作为研究对象,利用工程中常用的低阶数值格式,研究格式本身的数值误差对大涡模拟计算的影响.结果表明:三种数值格式的数值耗散行为都与亚格子模型行为类似,即在小雷诺数下,流场比较光滑时,耗散很小,当雷诺数增加,流动转捩为湍流,流场梯度增大,耗散显著增大.对于MUSCL格式和二阶有界中心格式,在高雷诺数下,亚格子尺度模型没有明显改善计算结果,但也没有使计算结果恶化.中心格式相比其它两种格式,数值耗散最小,但是在高雷诺数湍流情况下,中心格式的数值耗散仍然主导了能量的耗散,再添加亚格子模型,计算结果反而变得稍差.对于工程中的低阶格式而言,采用中心格式计算大涡模拟是比较好的选择,而且在计算不存在稳定性问题时,采用不添加亚格子模型的隐式大涡模拟效果更好.     

Effect of resonant-frequency mismatch on attractors

Resonant perturbations are effective for harnessing nonlinear oscillators for various applications such as controlling chaos and inducing chaos. Of physical interest is the effect of small frequency mismatch on the attractors of the underlying dynamical systems. By utilizing a prototype of nonlinear oscillators, the periodically forced Duffing oscillator and its variant, we find a phenomenon: resonant-frequency mismatch can result in attractors that are nonchaotic but are apparently strange in the sense that they possess a negative Lyapunov exponent but its information dimension measured using finite numerics assumes a fractional value. We call such attractors pseudo-strange. The transition to pesudo-strange attractors as a system parameter changes can be understood analytically by regarding the system as nonstationary and using the Melnikov function. Our results imply that pseudo-strange attractors are common in nonstationary dynamical systems.     

MHD from a Microscopic Concept and Onset of Turbulence in Hartmann Flow

We derive higher order magneto-hydrodynamic （MHD） equations from a microscopic picture using pro-jection and perturbation formalism. In an application to Hartmann flow we find velocity profiles flattening towards the center at the onset of turbulence in hydrodynamic limit. Comparison with the system under the effect of a uniform magnetic field yields difference in the onset of turbulence consistent with observations, showing that the presence of magnetic field inhibits onset of instability or turbulence. The laminar-turbulent transition is demonstrated in a phase transition plot of the development in time of the relative average velocities vs. Reynolds number showing a sharp increase of the relative average velocity at the transition point as determined by the critical Reynolds number.     

Novel four-dimensional autonomous chaotic system generating one-, two-, three- and four-wing attractors

In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincaré map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations are in good agreement with the numerical simulation results.     

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.     

Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation

This paper proposes a fractional-order simplest chaotic system using a bi-stable locally-active memristor. The characteristics of the memristor and transient transition behaviors of the proposed system are analyzed, and this circuit is implemented digitally using ARM-based MCU. Firstly, the mathematical model of the memristor is designed, which is nonvolatile, locally-active and bi-stable. Secondly, the asymptotical stability of the fractional-order memristive chaotic system is investigated and some sufficient conditions of the stability are obtained. Thirdly, complex dynamics of the novel system are analyzed using phase diagram, Lyapunov exponential spectrum, bifurcation diagram, basin of attractor, and coexisting bifurcation, coexisting attractors are observed. All of these results indicate that this simple system contains the abundant dynamic characteristics. Moreover, transient transition behaviors of the system are analyzed, and it is found that the behaviors of transient chaotic and transient period transition alternately occur. Finally, the hardware implementation of the fractional-order bi-stable locally-active memristive chaotic system using ARM-based STM32F750 is carried out to verify the numerical simulation results.     

二维logistic映射的动力学行为和奇怪吸引子的分形特征

研究了二维logistic映射的动力学行为和奇怪吸引子的分形特征.利用分岔图、相图和Lyapunov指数谱分析系统的分岔过程,研究系统通向混沌的道路并确定系统处于混沌运动的参数区间;采用G-P算法计算奇怪吸引子的关联维数和Kolmogorov熵,对奇怪吸引子的分形特征定量刻画;采用逃逸时间算法构造奇怪吸引子的彩色广义M-J集,对奇怪吸引子的分形特征定性表征.结果表明,这些分析方法的配合使用可以更全面、形象地描述奇怪吸引子的分形特征.     

Intermittency at the onset of stochasticity in nonlinear resonant coupling processes

A simple model for the nonlinear saturation of a plasma instability is studied via numerical solution of the resonant three-wave coupling equations. When parameters are varied the attractors of motion undergo bifurcations of several types. Intermittency is shown to occur in a transition from a limit cycle to a strange attractor. Such a transition might be indicative of an intermittent onset of turbulence in certain plasma experiments.     

具有自适应反馈突触的神经元模型中的混沌:电路设计

近期文献中报道了在具有自适应反馈突触的神经元模型中,随着参数的变化,存在从两个共存吸引子到一个相连吸引子再到两个共存吸引子的混沌转化现象.本文对此模型进行了电路设计,同时对具有非单调激活函数功能的电路设计进行了细致的研究,并利用Electronic Workbench (EWB)软件对所设计的电路进行了仿真实验,研究了电路中的混沌现象,验证了所设计电路的动力学行为与通过数值模拟结果十分相似. 关键词： 自适应反馈突触 神经元模型 混沌 电路设计     

Global aspects of turbulence induced by heteroclinic cycles in competitive diffusion Lotka-Volterra equation

This paper presents detailed numerical results of the competitive diffusion Lotka-Volterra equation (May-Leonard type). First, we derive the global phase diagrams of attractors in the parameter space including the system size, where transition lines between simple attractors are clearly obtained in accordance with the results of linear stability analysis, but the transition borders become complex when multi-basin structures appear. The complex aspects of the transition borders are studied in the case when the system size decreases. Next, we show the statistical aspects of the turbulence with special attention to the onset of the supercritical Hopf bifurcation. Several characteristic quantities, such as correlation length, correlation time, Lyapunov spectra and Lyapunov dimension, are investigated in detail near the onset of turbulence. Our data show the critical scaling law near the onset only in the restricted parameter domain. However even when the critical indices are not determined accurately, it is shown that the empirical scaling relations are obtained in a wide parameter domain far from the onset point and those scaling indices satisfy several relations. These scaling relations are discussed in comparison with the result derived by the phase reduction method. Lastly, we make a conjecture about the stability of an ecosystem based on the bifurcation diagram: the ecosystem obeying the Lotka-Volterra equation in the case of May-Leonard type is stabilized more as the system size increases.     

水力旋流器湍流流动的数值模拟

本文采用雷诺应力模型计算了水力旋流器的水相湍流流场,计算结果与实验数据吻合很好。与相关文献中采用修正的κ-ε模型的计算结果比较,本文采用雷诺应力模型的计算结果更接近于实验结果。计算得到了水力旋流器内的流线图、等压线以及零速包络面。     

Generalized dimensions,entropies, and Liapunov exponents from the pressure function for strange sets

For conformal mixing repellers such as Julia sets and nonlinear one-dimensional Cantor sets, we connect the pressure of a smooth transformation on the repeller with its generalized dimensions, entropies, and Liapunov exponents computed with respect to a set of equilibrium Gibbs measures. This allows us to compute the pressure by means of simple numerical algorithms. Our results are then extended to axiom-A attractors and to a nonhyperbolic invariant set of the line. In this last case, we show that a first-order phase transition appears in the pressure.     

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.     

Mixing in a T-shaped micromixer at moderate Reynolds numbers

In the present work, the regimes of the flow and mixing of fluids in a T-shaped micromixer in the range of the Reynolds numbers from 1 to 1000 are investigated systematically with the aid of numerical modeling. The flow and mixing regimes are shown to alter substantially with increasing Reynolds numbers. Five different flow regimes have been identified in the total. The dependencies of the friction coefficient and mixing efficiency on the Reynolds number are obtained. A sharp increase in the mixing efficiency at a flow transition from the symmetric to asymmetric steady regime is shown. On the other hand, the mixing efficiency slightly drops in the laminar-turbulent transition region. A substantial influence of the slip presence on walls on flow structure in the channel and mixing efficiency has been revealed.     

Simulation of laminar–turbulent transition in compressible Taylor–Green flow basing on quasi-gas dynamic equations

We report the results of numerical simulation of laminar–turbulent transition in the Taylor–Green vortex for viscous compressible gas flow basing on quasi-gas-dynamic (QGD) equations. Here the QGD system is obtained by a temporal averaging of the Navier–Stokes equations. The additional dissipative terms in QGD system serve to model the effects of the unresolved subgrid scales. Comparison with direct numerical simulation and large eddy simulation reference data demonstrates that QGD numerical algorithm provides a uniform and adequate simulation of both laminar and turbulent evolution of the vortex for Reynolds numbers from 100 up to 5000, including transition.     

The dynamical feature of transition of a Hamiltonian system to a dissipative system

The mechanism of generation and annihilation of attractors during transition from a Hamiltonian system to a dissipative system is studied numerically using the dissipative standard map. The transient process related to the formation of attracting basins of periodic attractors is studied by discussing the evolution of the KAM tori of the standard map. The result shows that as damping increases, attractors are mainly generated from elliptic orbits of the Hamiltonian system and annihilated by colliding with unstable periodic orbits originating from the corresponding hyperbolic orbits of the Hamiltonian system. The transient process also exhibits the general feature of bifurcation.     

Numerical simulation and flow visualization using soap film of the self-organized vortex structure in the wake of an array of cylinders

Abstract  

With the aid of computational fluid dynamics (CFD) and simple flow visualization technique using flowing soap-film, we present here the wake structures behind an array of cylinders for Reynolds numbers corresponding to both laminar and turbulent flow regimes. The image results illustrate interesting vortex interactions past these equally spaced cylinders; for low Reynolds number flow, well-organized wake pattern persists and manifests unsteadily to different symmetry states. An increase of free stream flow velocity causes the wake transition, resulting in the formation of asymmetric flow wake with chaotic mixing at the far wake. Observations from both the numerical simulations and soap-film are in good agreement at least qualitatively.