Stability analysis and rich oscillation patterns in discrete-time FitzHugh–Nagumo excitable systems with delayed coupling

In this paper, we give a detailed study of the stable region in discrete-time FitzHugh–Nagumo delayed excitable Systems, which can be divided into two parts: one is independent of delay and the other is dependent on delay. Two different new states are to be observed, which are new steady states (equilibria-the excitable FitzHugh–Nagumo) or limit cycles/higher periodic orbits (the FitzHugh–Nagumo oscillators) as the origin loses its stability, and usually, one is synchronized and the other asynchronized. We also find out that there exist critical curves through which there occur fold bifurcations, flip bifurcations, Neimark–Sacker bifurcations and even higher-codimensional bifurcations etc. It is also shown that delay can play an important role in rich dynamics, such as the occurrence of chaos or not, by means of Lyapunov exponents, Lyapunov dimensions, and the sensitivity to the initial conditions. Multistability phenomena are also discussed including the coexistence of synchronized and asynchronized oscillators, or synchronized/asynchronized oscillators and multiple stable nontrivial equilibria etc.     

Coherent neural oscillations induced by weak synaptic noise

Nonlinear Dynamics - We analyze the effect of synaptic noise on the dynamics of the FitzHugh–Nagumo (FHN) neuron model. In our deterministic parameter regime, a limit cycle solution cannot...     

Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures

This paper presents a three-dimensional autonomous Lorenz-like system formed by only five terms with a butterfly chaotic attractor. The dynamics of this new system is completely different from that in the Lorenz system family. This new chaotic system can display different dynamic behaviors such as periodic orbits, intermittency and chaos, which are numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation diagrams and Poincaré sections. Furthermore, this new system with compound structures is also proved by the presence of Hopf bifurcation at the equilibria and the crisis-induced intermittency.     

Non-lane-discipline-based car-following model incorporating the electronic throttle dynamics under connected environment

We investigate the nonlinear dynamics of a system of generalized Duffing-type MEMS resonator in the frame of simple analog electronic circuit. A mathematical model formed for the proposed generalized Duffing-type MEMS oscillator in which nonlinearities arising out of two different sources such as mid-plane stretching and electrostatic force can lead to variety of nonlinear phenomena such as period-doubling route, transient chaos and homo-/heteroclinic oscillations. These phenomena were confirmed through detailed numerical investigations such as phase portraits, bifurcation diagram, Poincaré map, Lyapunov exponent spectrum and finite-time Lyapunov exponent. The analog circuit realization for the Duffing-type MEMS resonator is constructed. The numerically simulated results are confirmed in the laboratory experimental observations which are closely matched with each other. The experimentally observed chaotic attractor confirmed through FFT spectrum, 0–1 test and Poincaré cross section. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio.     

Bursting generation mechanism in a three-dimensional autonomous system,chaos control,and synchronization in its fractional-order form

The bifurcation mechanism of bursting oscillations in a three-dimensional autonomous slow-fast Kingni et al. system (Nonlinear Dyn. 73, 1111–1123, 2013) and its fractional-order form are investigated in this paper. The stability analysis of the system is carried out assuming that the slow subsystem evolves on quasi-static state. It is reveaved that the bursting oscillations found in the system result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. We refer this class of bursting to “source/bursting.” The coexistence of symmetrical bursting limit cycles and chaotic bursting attractors is observed. In addition, the fractional-order chaotic slow-fast system is studied. The lowest order of the commensurate form of this system to exhibit chaotic behavior is found to be 2.199. By tuning the commensurate fractional-order, the chaotic slow-fast system displays Chen- and Lorenz-like chaotic attractors, respectively. The stability analysis of the controlled fractional-order-form of the system to its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Moreover, the synchronization of chaotic bursting oscillations in two identical fractional-order systems is numerically studied using the unidirectional linear error feedback coupling scheme. It is shown that the system can achieve synchronization for appropriate coupling strength. Furthermore, the effect of fractional derivatives orders on chaos control and synchronization is analyzed.     

Chaotic dynamics in a neural network under electromagnetic radiation

In this paper, a small Hopfield neural network with three neurons is studied, in which one of the three neurons is considered to be exposed to electromagnetic radiation. The effect of electromagnetic radiation is modeled and considered as magnetic flux across membrane of the neuron, which contributes to the formation of membrane potential, and a feedback with a memristive type is used to describe coupling between magnetic flux and membrane potential. With the electromagnetic radiation being considered, the previous steady neural network can present abundant chaotic dynamics. It is found that hidden attractors can be observed in the neural network under different conditions. Moreover, periodic motion and chaotic motion appear intermittently with variations in some system parameters. Particularly, coexistence of periodic attractor, quasiperiodic attractor, and chaotic strange attractor, coexistence of bifurcation modes and transient chaos can be observed. In addition, an electric circuit of the neural network is implemented in Pspice, and the experimental results agree well with the numerical ones.     

Reproduce the biophysical function of chemical synapse by using a memristive synapse

Dynamical modeling of nervous systems is of fundamental importance in many scientific fields containing the topics relative to computational neuroscience and biophysics. Many feasible mathematical models have been suggested in the explanation and prediction of some features of neural activities. Considering the special experimental findings and the computational efficiency, it is necessary to find a perfect balance between estimating biophysical functions with complete dynamics and reducing complexity when a tractable model is built. In this paper, a chemical synaptic model is reproduced by using a memristive synapse because it not only remains synaptic characteristic but also exhibits a pinched hysteresis loop and active feature locally. That is, a neuron activated by chemical synapse can produce similar firing modes as the neuron coupled by a memristive synapse, and both the chemical synapse and memristive synapse have similar field effect and biophysical properties. By calculating the one-parameter and two-parameter bifurcation as well as the Lyapunov exponent spectrum, it is confirmed that a neuron can be excited by the chemical synapse or the memristive synapse for generating chaotic firing patterns. Oscillation of the circuit composed of neuron and functional synapse is analyzed, suggesting that there exists a relation between the local activity and the edge of chaos via Hopf bifurcation. A modular circuit is designed to construct large-scale neural network. These results in this paper provide new evidences for application of memristive components and guide us to know the biophysical function of chemical synapse from physical viewpoint, in which the chemical synapse could be a kind of memristive synapse because of the same biophysical function.

     

Vibration energy harvesting under concurrent base and flow excitations with internal resonance

Nonlinear Dynamics - Modulational instability, as a mechanism of wave trains and soliton formation in biological system, is explored in the frame work of the new FitzHugh–Nagumo model. This...     

A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system

This paper presents a new four-dimensional (4-D) smooth quadratic autonomous chaotic system, which can present periodic orbit, chaos, and hyper-chaos under the conditions on different parameters. Importantly, the system can generate a four-wing hyper-chaotic attractor and a pair of coexistent double-wing hyper-chaotic attractors with two symmetrical initial conditions. Furthermore, a four-wing transient chaos occurs in the system. The dynamic analysis approach- in the paper involves time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, to investigate some basic dynamical behaviors of the proposed 4-D system.     

Bifurcation analysis of a two-DoF mechanical system subject to digital position control. Part II. Effects of asymmetry and transition to chaos

Considering a two DoF system subject to digital position control, of interest for robotic application, we analyze the dynamics of the system at the intersection of two loci of Neimark–Sacker bifurcations, where a double Neimark–Sacker bifurcation is taking place. In the system, the saturation of the control force is the only nonlinear term considered, other than this, the system is piecewise linear. Starting from the analytical investigation already performed in Part I (Habib et al. in Nonlin. Dyn., under review, 2013), in this paper the effects of an asymmetry of the saturation of the control force are investigated, both analytically and numerically. The results show the increasing complexity of the dynamics for a more and more asymmetric system. First, the asymmetry is making the bifurcation transit from supercritical to subcritical, then it generates a stable torus that breaks down into a strange attractor, associated with a chaotic motion. In the last part of the paper, the torus breakdown and the onset of chaos are investigated, furthermore the evolution of complex dynamics through regions of phase locking and higher-dimensional chaos is outlined.     

Generating tri-chaos attractors with three positive Lyapunov exponents in new four order system via linear coupling

This paper presents a new hyperchaotic system with three positive Lyapunov exponents (called Tri-Chaos). Via linear coupling, Mathieu, and van der Pol systems are coupled with each other and then become a new four order system??Mathieu?Cvan der Pol autonomous system. As we know, two positive Lyapunov exponents confirm hyperchaotic nature of its dynamics and it means that the system can present more complicated behavior than ordinary chaos. We further generate three positive Lyapunov exponents in a new coupled nonlinear system and anticipate the advanced application in secure communication. Not only a new chaotic system with three Lyapunov exponents is proposed, but also its implementation of an electronic circuit is put into practice in this article. The phase portrait, electronic circuit, power spectrum, Lyapunov exponents, and 2-D and 3-D parameter diagram of tri-chaos with three positive Lyapunov exponents of the new system will be shown in this paper.     

Synchronization of improved chaotic Colpitts oscillators using nonlinear feedback control

The model and the normalized state equations of the novel version of the Colpitts oscillator designed to operate in the ultra-high frequency range are presented. The circuit is investigated numerically and simulations demonstrate chaos in the microwave frequency range. Typical phase portrait, Lyapunov exponent and Lyapunov dimension are calculated using a piece-wise linear approximation of nonlinear I–V characteristic of the bipolar junction transistor. In addition, the feedback controller is applied to achieve chaos synchronization for two identical improved chaotic Colpitts oscillators. In the frame the nonlinear function of the system is used as a nonlinear feedback term for the stability of the error dynamics. Finally, numerical simulations show that this control method is feasible for this oscillator.     

The research on Cournot–Bertrand duopoly model with heterogeneous goods and its complex characteristics

This paper details the research of the Cournot–Bertrand duopoly model with the application of nonlinear dynamics theory. We analyze the stability of the fixed points by numerical simulation; from the result we found that there exists only one Nash equilibrium point. To recognize the chaotic behavior of the system, we give the bifurcation diagram and Lyapunov exponent spectrum along with the corresponding chaotic attractor. Our study finds that either the change of output modification speed or the change of price modification speed will cause the market to the chaotic state which is disadvantageous for both of the firms. The introduction of chaos control strategies can bring the market back to orderly competition. We exert control on the system with the application of the state feedback method and the parameter variation control method. The conclusion has great significance in theory innovation and practice.     

A priori estimates for solutions of FitzHugh–Rinzel system

Meccanica - The FitzHugh–Rinzel system is able to describe some biophysical phenomena, such as bursting oscillations, and the study of its solutions can help to better understand several...     

周期激励下广义蔡氏电路混沌运动中的概周期行为

由于广义蔡氏电路存在2个对称的稳定平衡点,周期激励可能导致系统出现相应于不同初值的2种共存的分岔模式. 概周期解由环面破裂进入混沌,混沌吸引子从相位不同步逐渐演化为同步,并进一步随着参数的变化,产生分裂现象. 分裂后的2个相互对称的混沌吸引子仍存在相位同步效应,这2个混沌吸引子再次相互作用后形成扩大了的混沌吸引子,并交替围绕2个子混沌结构来回振荡. 同时,在混沌过程中,其轨迹在相当长的一段时间内严格按照概周期行为振荡,即混沌结构中存在局部概周期行为,这种局部概周期行为随参数的变化会逐步减弱,直至消失.      

On the boundedness of solutions to the Lorenz-like family of chaotic systems

This paper deals with a class of three-dimensional autonomous nonlinear systems which have potential applications in secure communications, and investigates the localization problem of compact invariant sets of a class of Lorenz-like chaotic systems which contain T system with the help of iterative theorem and Lyapunov function theorem. Since the Lorenz-like chaotic system does not have y in the second equation, the approach used to the Lorenz system cannot be applied to the Lorenz-like chaotic system. We overcome this difficulty by introducing a cross term and get an interesting result, which includes the most interesting case of the chaotic attractor of the Lorenz-like systems. Furthermore, the results obtained in this paper are applied to study complete chaos synchronization. Finally, numerical simulations show the effectiveness of the proposed scheme.     

认知神经科学中蕴藏的力学思想与应用

该文系统总结了作者团队在脑科学领域内提出的神经能量理论与方法,以及力学与神经能量理论之间的内在联系.着重介绍了如何运用分析动力学的思想构建一个与H-H模型等效的W-Z神经元模型.并以此为基础,在神经科学领域内提出了以神经能量为核心的大尺度神经科学模型和大脑全局神经编码的理论框架.在包括视知觉等多个感知觉神经系统的信息处理、大脑的智力探索以及预测神经元新的工作机制、解释神经科学难以解释的实验现象等方面,证实了这个新颖的神经元模型所展现出来的独特功能与优势.由于可塑性是认知神经科学与智能行为的核心,通过蛋白质分子机器的经典力学分析,进一步阐明了神经元的可塑性和神经发育不仅仅只是生物化学反应过程,力学的作用与贡献也是不可或缺的重要因素.表明了力学科学在神经科学、生命科学中的研究思想及其内在逻辑的深远影响.这些研究对于今后推动实验神经科学与理论神经科学的融合,摒弃神经科学领域中还原论与整体论研究方法中的不足,并将它们各自的优点进行有效地整合,促进力学科学的理论与方法的渗透是极其重要的.      

A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–Nagumo model

This work presents a new method to calculate the Lyapunov spectrum of dynamical systems based on the time evolution of initially small disturbed copies (“clones”) of the motion equations. In this approach, it is not necessary to construct the tangent space associated with the time evolution of linearized versions of motion equations, being the Lyapunov exponents directly estimated in terms of the rate of convergence or divergence of these disturbed clones with respect to the fiducial trajectory, there being periodic correction via the Gram–Schmidt Reorthonormalization procedure. The proposed method offers the possibility of partial estimation of the Lyapunov spectrum and can also be applied to nonsmooth dynamics, since the linearization procedure is no longer required. The idea is tested for representative continuous- and discrete-time dynamical systems and validated by means of comparison with the classical method to perform this calculation. To illustrate its applicability in the nonsmooth context, the largest Lyapunov exponent of the FitzHugh–Nagumo neuronal model under discontinuous periodic excitation is calculated taking the amplitude of stimulation as control parameter. This analysis reveals some complex behaviours for this simple neuronal model, which motivates relevant discussions about the possible role of chaos in the cognitive process.     

Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system

This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time series, Lyapunov exponents’ spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the new system has strong hyperchaotic dynamics in itself. The Kaplan–Yorke dimension, Poincaré sections and the frequency spectra are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré–Andronov–Hopf bifurcation theorem. Numerical simulation results show consistency with our theoretical analysis.