A memristor oscillator based on a twin-T network

A novel inductance-free nonlinear oscillator circuit with a single bifurcation parameter is presented in this paper. This circuit is composed of a twin-T oscillator, a passive RC network, and a flux-controlled memristor. With an increase in the control parameter, the circuit exhibits complicated chaotic behaviors from double periodicity. The dynamic properties of the circuit are demonstrated by means of equilibrium stability, Lyapunov exponent spectra, and bifurcation diagrams. In order to confirm the occurrence of chaotic behavior in the circuit, an analog realization of the piecewise-linear flux-controlled memristor is proposed, and Pspice simulation is conducted on the resulting circuit.     

Effect of metal oxide arrester on the chaotic oscillations in the voltage transformer with nonlinear core loss model using chaos theory

In this paper, controlling chaos when chaotic ferroresonant oscillations occur in a voltage transformer with nonlinear core loss model is performed. The effect of a parallel metal oxide surge arrester on the ferroresonance oscillations of voltage transformers is studied. The metal oxide arrester(MOA) is found to be effective in reducing ferroresonance chaotic oscillations. Also the multiple scales method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes sub-harmonic, quasi-periodic, and also chaotic oscillations. In this paper, the chaotic behavior and various ferroresonant oscillation modes of the voltage transformer is studied. This phenomenon consists of different types of bifurcations such as period doubling bifurcation(PDB), saddle node bifurcation(SNB), Hopf bifurcation(HB), and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are illustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via the multiple scales method to obtain Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system.     

Codimension-two bifurcation of axial loaded beam bridge subjected to an infinite series of moving loads

A novel model is proposed which comprises of a beam bridge subjected to an axial load and an infinite series of moving loads. The moving loads, whose distance between the neighbouring ones is the length of the beam bridge, coupled with the axial force can lead the vibration of the beam bridge to codimension-two bifurcation. Of particular concern is a parameter regime where non-persistence set regions undergo a transition to persistence regions. The boundary of each stripe represents a bifurcation which can drive the system off a kind of dynamics and jump to another one, causing damage due to the resulting amplitude jumps. The Galerkin method, averaging method, invertible linear transformation, and near identity nonlinear transformations are used to obtain the universal unfolding for the codimension-two bifurcation of the mid-span deflection. The efficiency of the theoretical analysis obtained in this paper is verified via numerical simulations.     

Identification of Chaotic Systems with Application to Chaotic Communication

We propose and develop a novel method to identify a chaotic system with time-varying bifurcation parameters via an observation signal which has been contaminated by additive white Gaussian noise.This method is based on an adaptive algorithm,which takes advantage of the good approximation capability of the radial basis function neural network and the ability of the extended Kalman filter for tracking a time-varying dynamical system.It is demonstrated that,provided the bifurcation parameter varies slowly in a time window,a chaotic dynamical system can be tracked and identified continuously,and the time-varying bifurcation parameter can also be retrieved in a sub-window of time via a simple least-square-fit method.     

Effect of metal oxide arrester on the chaotic oscillations in the voltage transformer with nonlinear core loss model using chaos theory Effect of metal oxide arrester on the chaotic oscillations in the voltage transformer with nonlinear core loss model using chaos theory

In this paper, controlling chaos when chaotic ferroresonant oscillations occur in a voltage transformer with nonlin- ear core loss model is performed. The effect of a parallel metal oxide surge arrester on the ferroresonance oscillations of voltage transformers is studied. The metal oxide arrester （MOA） is found to be effective in reducing ferroresonance chaotic oscillations. Also the multiple scales method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes sub-harmonic, quasi-periodic, and also chaotic oscillations. In this paper, the chaotic behavior and various ferroresonant oscillation modes of the voltage transformer is studied. This phenomenon consists of different types of bifur- cations such as period doubling bifurcation （PDB）, saddle node bifurcation （SNB）, Hopf bifurcation （HB）, and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are il- lustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via the multiple scales method to obtain Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system.     

Bifurcation and chaos analysis of a nonlinear electromechanical coupling relative rotation system

Hopf bifurcation and chaos of a nonlinear electromechanical coupling relative rotation system are studied in this paper. Considering the energy in air-gap field of AC motor, the dynamical equation of nonlinear electromechanical coupling relative rotation system is deduced by using the dissipation Lagrange equation. Choosing the electromagnetic stiffness as a bifurcation parameter, the necessary and sufficient conditions of Hopf bifurcation are given, and the bifurcation characteristics are studied. The mechanism and conditions of system parameters for chaotic motions are investigated rigorously based on the Silnikov method, and the homoclinic orbit is found by using the undetermined coefficient method. Therefore, Smale horseshoe chaos occurs when electromagnetic stiffness changes. Numerical simulations are also given, which confirm the analytical results.     

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.     

Stochastic bifurcations of generalized Duffing–van der Pol system with fractional derivative under colored noise

The stochastic bifurcation of a generalized Duffing–van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.     

Evolution of entanglement between qubits ultra-strongly coupling to a quantum oscillator

We investigate the dynamics of two qubits coupled with a quantum oscillator by using the adiabatic approximation method. We take account of the interaction between the qubits and show how the entanglement is affected by the interaction parameter. The most interesting result is that we can prolong the entanglement time or improve the entanglement degree by using an appropriate interaction parameter. As the generation and preservation of entanglement of qubits are crucial for quantum information processing, our research will be useful.     

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.     

有界随机噪声激励下软弹簧Duffing振子的安全盆分叉

研究了软弹簧Duffing振子在有界随机噪声激励下，系统安全盆的侵蚀现象，并提出了随机安全盆分叉的概念. 计算表明，由于随机扰动的影响，系统的随机安全盆分叉点发生了偏移. 关键词： Duffing振子 有界随机噪声 安全盆 分叉     

一类Hopf分岔系统的通用鲁棒稳定控制器设计方法

针对一类多项式形式的Hopf分岔系统, 提出了一种鲁棒稳定的控制器设计方法. 使用该方法设计控制器时不需要求解出系统在分岔点处的分岔参数值, 只需要估算出分岔参数的上下界, 然后设计一个参数化的控制器, 并通过Hurwitz判据和柱形代数剖分技术求解出满足上下界条件的控制器参数区域, 最后在得到的这个区域内确定出满足鲁棒稳定的控制器参数值. 该方法设计的控制器是由包含系统状态的多项式构成, 形式简单, 具有通用性, 且添加控制器后不会改变原系统平衡点的位置. 本文首先以Lorenz系统为例说明了控制器的推导和设计过程, 然后以van der Pol振荡系统为例, 进行了工程应用. 通过对这两个系统的控制器设计和仿真, 说明了文中提出的控制器设计方法能够有效地应用于这类Hopf分岔系统的鲁棒稳定控制, 并且具有通用性.     

On the stationary distribution of self-sustained oscillators around bifurcation points

A double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones. Systems undergoing Hopf bifurcations belong to this class. As an illustrative example, the generalized van der Pol oscillator is considered around its bifurcation point. A calculation is carried out up to third order in both the noise intensity and the bifurcation parameter (damping coefficient).     

Heteroclinic Bifurcation of Strongly Nonlinear Oscillator

Analytical prediction of heteroclinic bifurcation of the strongly nonlinear oscillator is presented by using the extended normal form method. We consider the approximate periodic solution of the system subject to the quintic nonlinearity by introducing the undetermined fundamental frequency. For the occurrence of heteroclinicity, the bifurcation criterion is accomplished. It depends on the contact of the limit cycle with the saddle equilibrium. As is illustrated, the explicit application shows that the new results coincide very well with the results of numerical simulation when disturbing parameter is of arbitrary magnitude.     

High-codimensional static bifurcations of strongly nonlinear oscillator

The static bifurcation of the parametrically excited strongly nonlinear oscillator is studied. We consider the averaged equations of a system subject to Duffing--van der Pol and quintic strong nonlinearity by introducing the undetermined fundamental frequency into the computation in the complex normal form. To discuss the static bifurcation, the bifurcation problem is described as a 3-codimensional unfolding with $Z_{2}$ symmetry on the basis of singularity theory. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied by using the eigenvalues in various parameter regions.     

单结晶体管二阶RLC串联电路中的周期分岔和混乱运动

本文提出一种新的非线性二阶单结晶体管强迫振荡电路,能在周期激励项参数变动的很大范围内发生倍周期分岔和混乱运动现象。 关键词：     

Period-doubling cascades of canards from the extended Bonhoeffer-van der Pol oscillator

This Letter investigates the period-doubling cascades of canards, generated in the extended Bonhoeffer-van der Pol oscillator. Canards appear by Andronov-Hopf bifurcations (AHBs) and it is confirmed that these AHBs are always supercritical in our system. The cascades of period-doubling bifurcation are followed by mixed-mode oscillations. The detailed two-parameter bifurcation diagrams are derived, and it is clarified that the period-doubling bifurcations arise from a narrow parameter value range at which the original canard in the non-extended equation is observed.     

窄带随机噪声作用下单自由度非线性干摩擦系统的响应

研究了单自由度非线性干摩擦系统在窄带随机噪声参数激励下的主共振响应问题.用Krylov-Bogoliubov平均法得到了关于慢变量的随机微分方程.在没有随机扰动情形,得到了系统响应幅值满足的代数方程.在有随机扰动情形,用线性化方法和矩方法给出了系统响应稳态矩计算的近似计算公式.讨论了系统阻尼项、非线性项、随机扰动项和干摩擦项等参数对于系统响应的影响.理论计算和数值模拟表明,当非线性强度增大时系统的响应显著变小,系统分岔点滞后;随着激励频率的增大系统响应变大,而当激励频率小于一定的值时,系统响应为零;增加干 关键词： 单自由度非线性干摩擦系统 主共振响应 Krylov-Bogoliubov平均法