Synchronization transition of a coupled system composed of neurons with coexisting behaviors near a Hopf bifurcation

The coexistence of a resting condition and period-1 firing near a subcritical Hopf bifurcation point, lying between the monostable resting condition and period-1 firing, is often observed in neurons of the central nervous systems. Near such a bifurcation point in the Morris-Lecar （ML） model, the attraction domain of the resting condition decreases while that of the coexisting period-1 firing increases as the bifurcation parameter value increases. With the increase of the coupling strength, and parameter and initial value dependent synchronization transition processes from non-synchronization to compete synchronization are simulated in two coupled ML neurons with coexisting behaviors： one neuron chosen as the resting condition and the other the coexisting period-1 firing. The complete synchronization is either a resting condition or period-1 firing dependent on the initial values of period-1 firing when the bifurcation parameter value is small or middle and is period- 1 firing when the parameter value is large. As the bifurcation parameter value increases, the probability of the initial values of a period- 1 firing neuron that lead to complete synchronization of period- 1 firing increases, while that leading to complete synchronization of the resting condition decreases. It shows that the attraction domain of a coexisting behavior is larger, the probability of initial values leading to complete synchronization of this behavior is higher. The bifurcations of the coupled system are investigated and discussed. The results reveal the complex dynamics of synchronization behaviors of the coupled system composed of neurons with the coexisting resting condition and period-1 firing, and are helpful to further identify the dynamics of the spatiotemporal behaviors of the central nervous system.     

Noise-induced synchronous stochastic oscillations small scale cultured heart-cell networks

This paper reports that the synchronous integer multiple oscillations of heart-cell networks or clusters are observed in the biology experiment. The behaviour of the integer multiple rhythm is a transition between super- and sub- threshold oscillations, the stochastic mechanism of the transition is identified. The similar synchronized oscillations are theoretically reproduced in the stochastic network composed of heterogeneous cells whose behaviours are chosen as excitable or oscillatory states near a Hopf bifurcation point. The parameter regions of coupling strength and noise density that the complex oscillatory rhythms can be simulated are identified. The results show that the rhythm results from a simple stochastic alternating process between super- and sub-threshold oscillations. Studies on single heart cells forming these clusters reveal excitable or oscillatory state nearby a Hopf bifurcation point underpinning the stochastic alternation. In discussion, the results are related to some abnormal heartbeat rhythms such as the sinus arrest.     

Hopf bifurcation analysis and circuit implementation for a novelfour-wing hyper-chaotic system

In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare′ maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.     

Control of Codimension-2 Bautin Bifurcation in Chaotic Lu System

In this paper, we design a feedback controller, and analytically determine a control criterion so as to control the codimension-2 Bautin bifurcation in the chaotic Lfi system. According to the control criterion, we determine a potential Bautin bifurcation region （denoted by P） of the controlled system. This region contains the Bautin bifurcation region （denoted by Q） of the uncontrolled system as its proper subregion. The controlled system can exhibit Bautin bifurcation in P or its proper subregion provided the control gains are properly chosen. Particularly, we can control the appearance of Bautin bifurcation at any appointed point in the region P. Due to the relationship between Bantin bifurcation and Hopf bifurcation, the control scheme thereby is also viable for controlling the creation and stability of the Hopf bifurcation. In the controller, there are two terms： a linear term and a nonlinear cubic term. We show that the former determines the location of the Hopf bifurcation while the latter regulates its criticality. We also carry out numerical studies, and the simulation results confirm our analyticai predictions.     

Nonlinear dynamic characteristics and optimal control of a giant magnetostrictive film-shaped memory alloy composite plate subjected to in-plane stochastic excitation

The nonlinear dynamic characteristics and optimal control of a giant magnetostrictive film(GMF)-shaped memory alloy(SMA) composite plate subjected to in-plane stochastic excitation are studied. GMF is prepared based on an SMA plate, and combined into a GMF–SMA composite plate. The Van der Pol item is improved to explain the hysteretic phenomena of GMF and SMA, and the nonlinear dynamics model of a GMF–SMA composite cantilever plate subjected to in-plane stochastic excitation is developed. The stochastic stability of the system is analyzed, and the steady-state probability density function of the dynamic response of the system is obtained. The condition of stochastic Hopf bifurcation is discussed, the reliability function of the system is provided, and then the probability density of the first-passage time is given. Finally, the stochastic optimal control strategy is proposed by the stochastic dynamic programming method.Numerical simulation shows that the stability of the trivial solution varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the system’s reliability is improved through stochastic optimal control, and the firstpassage time is delayed. A GMF–SMA composite plate combines the advantages of GMF and SMA, and can reduce vibration through passive control and active control effectively. The results are helpful for the engineering applications of GMF–SMA composite plates.     

Dynamical investigation and parameter stability region analysis of a flywheel energy storage system in charging mode

In this paper, the dynamic behavior analysis of the electromechanical coupling characteristics of a flywheel energy storage system （FESS） with a permanent magnet （PM） brushless direct-current （DC） motor （BLDCM） is studied. The Hopf bifurcation theory and nonlinear methods are used to investigate the generation process and mechanism of the coupled dynamic behavior for the average current controlled FESS in the charging mode. First, the universal nonlinear dynamic model of the FESS based on the BLDCM is derived. Then, for a 0.01 kWh/1.6 kW FESS platform in the Key Laboratory of the Smart Grid at Tianjin University, the phase trajectory of the FESS from a stable state towards chaos is presented using numerical and stroboscopic methods, and all dynamic behaviors of the system in this process are captured. The characteristics of the low-frequency oscillation and the mechanism of the Hopf bifurcation are investigated based on the Routh stability criterion and nonlinear dynamic theory. It is shown that the Hopf bifurcation is directly due to the loss of control over the inductor current, which is caused by the system control parameters exceeding certain ranges. This coupling nonlinear process of the FESS affects the stability of the motor running and the efficiency of energy transfer. In this paper, we investigate into the effects of control parameter change on the stability and the stability regions of these parameters based on the averaged-model approach. Furthermore, the effect of the quantization error in the digital control system is considered to modify the stability regions of the control parameters. Finally, these theoretical results are verified through platform experiments.     

A New Route to the Interpretation of Hopf Invariant

We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the Ф-mapping topological current theory. The main purpose of this paper is to present a new theoretical framework, which can directly give the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclidean space R^2n-1. For the sake of this purpose we introduce a topological tensor current, which can naturally deduce the （n- 1）-dimensional topological defect in R^2n-1 space. If these （n- 1）-dimensional topological defects are closed oriented submanifolds of R^2n-1, they are just the （n - 1）-dimensional knots. The linking number of these knots is well defined. Using the inner structure of the topological tensor current, the relationship between Hopf invariant and the linking numbers of the higher-dimensional knots can be constructed.     

Topological Aspects of Solitons in Ferromagnets

Two kinds of topological soliton （skyrmion and magnetic vortex ring） in ferromagnets are studied. They have the common topological origin, a tensor Hαβ = n→·（δαn→×βδn→）, which describes the non-trivial distribution of local orientation of magnetization n→ at large distances in space. The topological stability of skyrmion is protected by the winding number. Knot-like topological defect as magnetic vortex rings is also studied. On the assumption that magnetic vortex rings are geometric lines, we present their δ-function distribution in ferromagnetic materials. Furthermore, it is briefly shown that Hopf invariant is a proper topological invariant to describe the topology of magnetic vortex rings.     

Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a new hyperchaotic circuit system. The stability of the hyperchaotie circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. An important feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.     

Knotted Topological Phase Singularities of Electromagnetic Field

In this paper, knotted objects （RS vortices） in the theory of topological phase singularity in electromagnetic field have been investigated in details. By using the Duan＇s topological current theory, we rewrite the topological current form of RS vortices and use this topological current we reveal that the Hopf invariant of RS vortices is just the sum of the linking and self-linking numbers of the knotted RS vortices. Furthermore, the conservation of the Hopf invariant in the splitting, the mergence and the intersection processes of knotted RS vortices is also discussed.     

Topology of Knotted Optical Vortices

Optical vortices as topological objects exist ubiquitously in nature. In this paper, by making use of the Duan＇s topological current theory, we investigate the topology in the closed and knotted optical vortices. The topological inner structure of the optical vortices are obtained, and the linking of the knotted optical vortices is also given.     

Topological Effect of Reduced Quantum Trajectory in Coherence and Decoherence

The topological expression of the reduced quantum trajectory is given by using C-mapping topological current theory. The topological expression is used in study of the coherence and the decoherence. We find the expression of vorticity for decoherence has a similar form as that for coherence. A topological reason leading to the decoherence is given and a new parameter is defined to indicate the coherence degree. The parameter is different from the damping factor because it relates to the topological structure of the reduced quantum trajectory.     

Estimating Topology of Discrete Dynamical Networks

In this paper, by applying Lasalle＇s invariance principle and some results about the trace of a matrix, we propose a method for estimating the topological structure of a discrete dynamical network based on the dynamical evolution of the network. The network concerned can be directed or undirected, weighted or unweighted, and the local dynamics of each node can be nonidentical. The connections among the nodes can be all unknown or partially known, Finally, two examples, including a Henon map and a central network, are illustrated to verify the theoretical results.     

Vortex flow in freestanding smectic films driven by elastic relaxation of the c director

We report a simple experiment in freestanding smectic films in which elastic distortions of the c director drive macroscopic flow. The flow field is visualized with tracer particles. Measurements are compared to predictions of a model that employs the coupled dynamic equations for director and velocity fields. Relaxation dynamics depends on the topology of the film center: for defect-free target patterns, shear flow provides the dominating contribution to the c director dynamics. In presence of a central topological defect of strength S = + 1, the influence of flow on the relaxation dynamics is practically negligible, while for a central S = - 1 defect, the influence of vortex flow on the c-director relaxation is roughly twice as large as for the defect-free state.     

Topological textures and their bifurcation processes in 2D ferromagnetic thin films

In this paper, by the use of the topological current theory, the topological structures and the dynamic processes in thin-film ferromagnetic systems are investigated directly from the viewpoint of topology. It is found that the topological charge of a thin-film ferromagnetic system can be changed by annihilation or creation processes of opposite polarized vortex–antivortex pairs taking place at space–time singularities of the normalized magnetization vector field of the system, the variation of the topological charge is integer and can further be expressed in terms of the Hopf indices and Brouwer degrees of the magnetization vector field around the singularities. Moreover, the change of the topological charge of the system is crucial to vortex core reversal processes in ferromagnetic thin films. With the help of the topological current theory and implicit function theorem, the processes of vortex merging, splitting as well as vortex core reversal are discussed in detail.     

数字全息术测定涡旋光束拓扑电荷数

提出了一种基于数字全息技术测定涡旋光束拓扑电荷数的方法.该方法通过数字全息技术获取涡旋光束和参考光的全息图并重构出涡旋光束的波前相位,判定相位围绕相位奇点的周期性分布来测定涡旋光束的拓扑电荷数.在拓扑电荷数取值分别为整数和分数的情况下,通过对数值模拟和实验结果的比较,表明该方法能够较准确地测定出拓扑电荷数.     

Fermi surface topology and the upper critical field in two-band superconductors: application to MgB2

Recent measurements of the anisotropy of the upper critical field B(c2) on MgB2 single crystals have shown a puzzling strong temperature dependence. Here, we present a calculation of the upper critical field based on a detailed modeling of band structure calculations that takes into account both the unusual Fermi surface topology and the two gap nature of the superconducting order parameter. Our results show that the strong temperature dependence of the B(c2) anisotropy can be understood as an interplay of the dominating gap on the sigma band, which possesses a small c-axis component of the Fermi velocity, with the induced superconductivity on the pi-band possessing a large c-axis component of the Fermi velocity. We provide analytic formulas for the anisotropy ratio at T=0 and T=T(c) and quantitatively predict the distortion of the vortex lattice based on our calculations.     

基于VLES模型的振动圆柱数值模拟

本文将一种VLES(Very Large Eddy Simulation)模型引入到动网格数值计算中,并验证了VLES模型用于模拟类似振动圆柱绕流的动边界问题的有效性.数值求解了不同振幅和频率下非稳态振动圆柱绕流问题.研究表明:随着振幅和激励频率的增加,绕圆柱流动涡脱离形式从2S模式转换到2P0模式,再到P+S模式.在...     

Hopf Bifurcations,Drops in the Lid-Driven Square Cavity Flow

The lid-driven square cavity flow is investigated by numerical experiments. It is found that from $ \mathrm{Re} $$=$ $5,000 $ to $ \mathrm{Re} $$=$$ 7,307.75 $ the solution is stationary, but at $ \mathrm{Re}$$=$$7,308 $ the solution is time periodic. So the critical Reynolds number for the first Hopf bifurcation localizes between $ \mathrm{Re} $$=$$ 7,307.75 $ and $ \mathrm{Re} $$=$$ 7,308 $. Time periodical behavior begins smoothly, imperceptibly at the bottom left corner at a tiny tertiary vortex; all other vortices stay still, and then it spreads to the three relevant corners of the square cavity so that all small vortices at all levels move periodically. The primary vortex stays still. At $ \mathrm{Re} $$=$$ 13,393.5 $ the solution is time periodic; the long-term integration carried out past $ t_{\infty} $$=$$ 126,562.5 $ and the fluctuations of the kinetic energy look periodic except slight defects. However, at $ \mathrm{Re} $$=$$ 13,393.75 $ the solution is not time periodic anymore: losing unambiguously, abruptly time periodicity, it becomes chaotic. So the critical Reynolds number for the second Hopf bifurcation localizes between $ \mathrm{Re} $$=$$ 13,393.5 $ and $ \mathrm{Re} $$=$$ 13,393.75 $. At high Reynolds numbers $ \mathrm{Re} $$=$$ 20,000 $ until $ \mathrm{Re} $$=$$ 30,000 $ the solution becomes chaotic. The long-term integration is carried out past the long time $ t_{\infty} $$=$$ 150,000 $, expecting the time asymptotic regime of the flow has been reached. The distinctive feature of the flow is then the appearance of drops: tiny portions of fluid produced by splitting of a secondary vortex, becoming loose and then fading away or being absorbed by another secondary vortex promptly. At $ \mathrm{Re} $$=$$ 30,000 $ another phenomenon arises—the abrupt appearance at the bottom left corner of a tiny secondary vortex, not produced by splitting of a secondary vortex.