Dynamics and stability of a new class of periodic solutions of the optical parametric oscillator

The stability and dynamics of a new class of periodic solutions is investigated when a degenerate optical parametric oscillator system is forced by an external pumping field with a periodic spatial profile modeled by Jacobi elliptic functions. Both sinusoidal behavior as well as localized hyperbolic (front and pulse) behavior can be considered in this model. The stability and bifurcation behaviors of these transverse electromagnetic structures are studied numerically. The periodic solutions are shown to be stabilized by the nonlinear parametric interaction between the pump and signal fields interacting with the cavity diffraction, attenuation, and periodic external pumping. Specifically, sinusoidal solutions result in robust and stable configurations while well-separated and more localized field structures often undergo bifurcation to new steady-state solutions having the same period as the external forcing. Extensive numerical simulations and studies of the solutions are provided.     

A flux-controlled model of meminductor and its application in chaotic oscillator

A meminductor is a new type of memory device developed from the memristor.We present a mathematical model of a flux-controlled meminductor and its equivalent circuit model for exploring the properties of the meminductor in a nonlinear circuit.We explore the response characteristics of the meminductor under the exciting signals of sinusoidal,square,and triangular waves by using theoretical analysis and experimental tests,and design a meminductor-based oscillator based on the model.Theoretical analysis and experiments show that the meminductor-based oscillator possesses complex bifurcation behaviors and can generate periodic and chaotic oscillations.A special phenomenon called the co-existent oscillation that can generate multiple oscillations(such as chaotic,periodic oscillations as well as stable equilibrium) with the same parameters and different initial conditions occurs.We also design an analog circuit to realize the meminductor-based oscillator,and the circuit experiment results are in accordance with the theory analysis.     

分数阶时滞反馈对Duffing振子动力学特性的影响

研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.     

一类具时滞的厄尔尼诺-南方涛动充电-放电振子模型的Hopf分岔与周期解问题

研究了一类具时滞的厄尔尼诺-南方涛动充电-放电振子模型. 给出了产生Hopf分岔的临界时滞条件, 利用Mawhin重合度理论,探讨了该模型的周期解问题. 关键词： 时滞 厄尔尼诺-南方涛动模型 Hopf分岔 周期解     

Three-dimensional chaotic autonomous van der pol–duffing type oscillator and its fractional-order form

In this paper, a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator is proposed. The three-dimensional autonomous VdPD oscillator is obtained by replacing the external periodic drive source of two-dimensional chaotic nonautonomous VdPD type oscillator by a direct positive feedback loop. By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established. The dynamical properties of proposed three-dimensional autonomous VdPD type oscillator is investigated showing that for a suitable choice of the parameters, it can exhibit periodic behaviors, chaotic behaviors and coexistence between periodic and chaotic attractors. Moreover, the physical existence of the chaotic behavior and coexisting attractors found in three-dimensional proposed autonomous VdPD type oscillator is verified by using Orcard-PSpice software. A good qualitative agreement is shown between the numerical simulations and Orcard-PSpice results. In addition, fractional-order chaotic three-dimensional proposed autonomous VdPD type oscillator is studied. The lowest order of the commensurate form of this oscillator to exhibit chaotic behavior is found to be 2.979. The stability analysis of the controlled fractional-order-form of proposed three-dimensional autonomous VdPD type oscillator at its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Finally, an example of observer-based synchronization applied to unidirectional coupled identical proposed chaotic fractional-order oscillator is illustrated. It is shown that synchronization can be achieved for appropriate coupling strength.     

The dynamics of the pendulum suspended on the forced Duffing oscillator

We investigate the dynamics of the pendulum suspended on the forced Duffing oscillator. The detailed bifurcation analysis in two parameter space (amplitude and frequency of excitation) which presents both oscillating and rotating periodic solutions of the pendulum has been performed. We identify the areas with low number of coexisting attractors in the parameter space as the coexistence of different attractors has a significant impact on the practical usage of the proposed system as a tuned mass absorber.     

Bifurcations in a system of two identical diffusion-coupled relaxational brusselators

We propose a two-parameter bifurcation analysis of the dynamics of a system of two identical asymmetrically coupled Brusselators. The stability boundaries of the inhomogeneous steady states and periodic attractors are calculated as the functions of the constraint force and one of the free parameters. The coexistence of different attractors giving rise to multirhythmicity of the dynamics is studied as well as the bifurcation transitions between them. It is shown that the relaxation ability of an oscillator plays an important role in the simplification of the phase diagram, since it removes the overlapping of the existence region for different solutions. We assume that the results primarily characterize the properties of the diffusion coupling and, therefore, they can be applied in the study of other oscillator systems.P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 5, pp. 373–401, May, 1995.     

周期切换下Chen系统的振荡行为与非光滑分岔分析

研究了不同参数Chen系统之间进行周期切换时的分岔和混沌行为.基于平衡态分析,考虑Chen系统在不同稳态解时通过周期切换连接生成的复合系统的分岔特性,得到系统的不同周期振荡行为.在演化过程中,由于切换导致的非光滑性,复合系统不仅仅表现为两子系统动力特性的简单连接,而且会产生各种分岔,导致诸如混沌等复杂振荡行为.通过Poincaré映射方法,讨论了如何求周期切换系统的不动点和Floquet特征乘子.基于Floquet理论,判定系统的周期解是渐近稳定的.同时得到,随着参数变化,系统既可以由倍周期分岔序列进入混沌,也可以由周期解经过鞍结分岔直接到达混沌.研究结果揭示了周期切换系统的非光滑分岔机理.     

非线性切换系统的动力学行为分析

建立了周期切换下的非线性电路模型,基于子系统平衡点及其稳定性分析,分别给出了其相应的fold分岔和Hopf分岔条件,讨论了子系统在不同平衡态下由周期切换导致的各种复杂行为,指出切换系统的周期解随参数的变化存在着倍周期分岔和鞍结分岔两种失稳情形,并相应地导致不同的混沌振荡,进而结合系统轨迹及其相应的分岔分析,揭示了各种振荡模式的动力学机理. 关键词： 周期切换 倍周期分岔 鞍结分岔 混沌     

Duffing振子系统周期解的唯一性与精确周期信号的获取方法

研究Duffing振子系统的周期解的唯一性与精确周期信号的获取方法.应用定性分析方法,获得了一类Duffing振子系统具有唯一周期解的必要条件,同时也得到了一类更广泛的非线性周期系统的周期解的唯一性.在一定条件下,给出了Duffing振子系统精确周期信号的获取方法。     

参外联合激励复合非线性振子的分岔分析

讨论了参外联合激励复合非线性振子的动力学行为.对其定常解在一阶近似下的方程进行了局部分岔分析,给出了简单分岔和Hopf分岔发生的条件,并通过对近似方程和原系统的数值模拟加以验证.分析了多种参数对该振子动力学行为演化过程的影响.根据全局分岔理论探讨了该振子在不同条件下发生同宿、异宿分岔的必要条件,其结论与数值计算的结果大致符合. 关键词： 复合非线性振子 局部分岔 全局分岔 混沌     

计算相位响应曲线的方波扰动直接算法

通过相位响应曲线可对具有极限环周期运动的动力系统的性质有更为深入的理解.神经元是一个典型的动力系统,因此相位响应曲线提供了一种研究神经元重复周期放电行为的新思路.本文提出一种求解相位响应曲线的方法,即方波扰动的直接算法,通过Hodgkin-Huxley,Fitz Hugh-Nagumo,Morris-Lecar和Hindmarsh-Rose神经元模型验证该算法可计算周期峰放电、周期簇放电的相位响应曲线.该算法克服了其他算法在运用过程中的局限性.利用该算法计算结果表明:周期峰放电的相位响应曲线类型是由其分岔类型所决定;在Morris-Lecar模型中发现一种开始于Hopf分岔终止于鞍点同宿轨道分岔的阈上周期振荡,其相位响应曲线属于第二类型.通过大量的相位响应曲线的计算发现相位响应的相对大小及正负性仅取决于扰动所施加的时间,而且周期簇放电的相位响应曲线比周期峰放电的相位响应曲线更为复杂.     

Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method

In this paper, an explicit formulation of the incremental harmonic balance (IHB) scheme for computation of periodic solutions of a harmonically excited oscillator which is asymmetric with both stiffness and viscous damping piecewise linearities is derived. Analysis of dynamical behavior as bifurcation and chaos of the non-linear vibration system considered is effectively carried out by the IHB procedure, showing that the system exhibits chaos via the route of period-doubling bifurcation, with coexistence of multiple periodic attractors observed and analyzed by the interpolated cell mapping method. In addition, numerical simulation by the IHB method is compared with that by the fourth order Runge-Kutta numerical integration routine, which shows that this method is in many respects distinctively advantageous over classical approaches, and especially excels in performing parametric studies.     

Nonlinear oscillations of the FitzHugh-Nagumo equations under combined external and two-frequency parametric excitations

The continuous FitzHugh-Nagumo (FHN for short) model is transformed into modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations. At the first, the dependence of the solutions on a combined external and two-frequency parametric stimulus forcing is investigated. By using the multiple scale method, ranges of applied current and/or parametric forcing in which nonlinear oscillations are observed are described. Second, when the multiple scale method cannot be used, we numerically prove that in the modified van der Pol oscillator with asymmetry under external and two-frequency parametric excitations, chaos and periodic solution depending on the combination between different frequencies of the model should appear. We also show that the amplitude of the oscillations can be reduced or increased. To do this, we perform the study of the FHN model by choosing a range of parameters exhibiting Hopf bifurcation and two qualitative different regimes in phase portrait.     

周期切换下Rayleigh振子的振荡行为及机理

本文研究了自治与非自治电路系统在周期切换连接下的动力学行为及机理.基于自治子系统平衡点和极限环的相应稳定性分析和切换系统李雅普诺夫指数的理论推导及数值计算.讨论了两子系统在不同参数下的稳态解在周期切换连接下的复合系统的各种周期振荡行为,进而给出了切换系统随参数变化下的最大李雅普诺夫指数图及相应的分岔图,得到了切换系统在不同参数下呈现出周期振荡,概周期振荡和混沌振荡相互交替出现的复杂动力学行为并分析了其振荡机理.给出了切换系统通过倍周期分岔,鞍结分岔以及环面分岔到达混沌的不同动力学演化过程.     

二维抛物线离散映射的动力学研究

由两个一维抛物线离散映射作推广并非线性耦合,实现了一个新的二维抛物线离散映射.利用不动点稳定性分析和映射分岔分析,研究了所提出的二维离散映射的复杂动力学行为及其吸引子的演变过程,阐述了它所特有的共存分岔模式和快慢周期振荡效应等动力学特性.研究结果表明:二维抛物线离散映射具有动力学特性调节和动态幅度调节的两个功能不同的控制参数,存在Hopf分岔、分岔模式共存、锁频和周期振荡快慢效应等非线性物理现象.并基于微控制器实现的数字电路验证了相应的理论分析和数值仿真结果. 关键词： 二维离散映射 分岔 吸引子 参数     

分段线性电路切换系统的复杂行为及非光滑分岔机理

分析了分段线性电路系统在周期切换下的复杂动力学行为及其产生的机理. 基于平衡点分析, 给出了两子系统Fold分岔和Hopf分岔条件. 考虑了在不同稳定态时两子系统周期切换的分岔特性, 产生了不同的周期振荡, 并揭示了其产生的机理. 在不同的周期振荡中, 切换点的数量随参数变化产生倍化, 导致切换系统由倍周期分岔进入混沌. 关键词： 分段线性电路 切换系统 非光滑分岔     

Infinitely many coexisting attractors of a dual memristive Shinriki oscillator and its FPGA digital implementation

In this paper, a novel chaotic oscillator is proposed, which is derived from the classical Shinriki oscillator by substituting the series-parallel diode loop with a flux-controlled memristor and connecting an active charge-controlled memristor in series with an inductor. The mathematical model of the circuit is established, and the stability distribution maps of three non-zero eigenvalues in the equilibrium plane are obtained. The basic dynamical behaviors depending on the variation of the circuit parameters and memristor initial conditions are investigated by standard nonlinear analysis tools, such as bifurcation diagrams, Lyapunov exponents and phase portraits. Particularly, some striking phenomena, including the routes to double-scroll chaotic attractors, coexisting periodic-chaotic bubbles and asymmetric coexisting behaviors are observed. Furthermore, extreme multistability of the new oscillator is revealed by attraction basins under the initial condition of different dynamic elements. Finally, the Shinriki oscillator with two memristors is realized through Field-Programmable Gate Array (i.e., FPGA) to verify the effectiveness of the numerical simulations.     

用连续法计算五维对流模型的定常解和周期解

利用连续算法(Continuation algorithm)对五维对流非线性动力系统的定常解和周期解进行了数值计算。在参数平面Ri-Re上计算出实分岔点曲线、极限点曲线、Hopf分岔点曲线,绘出了分岔图。在分岔图上的不同区域,存在性质不同的稳定解如定常吸引子、周期吸引子等。分析了定常解、周期解的分岔过程。计算结果很好地说明大气中由基本态到对流态再到波动态最后到湍流态的物理转换过程。 连续算法对研究非线性动力系统的分岔以及耗散结构是很有效的计算方法。     

Dynamical behaviors of a system with switches between the Rössler oscillator and Chua circuits

The behaviors of a system that alternates between the Rössler oscillator and Chua's circuit is investigated to explore the influence of the switches on the dynamical evolution. Switches related to the state variables are introduced, upon which a typical switching dynamical model is established. Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points, which divide the parameters into several regions corresponding to different types of attractors. The dynamics behave typically in period orbits with the variation of the parameters. The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement. The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches. Furthermore, period-decreasing sequences have been obtained, which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.