基于可视化计算的非线性Lienard方程仿真研究

该文研究非线性Ｌｉｅｎａｒｄ方程的图形建模及可视化仿真问题。在图形环境下对Ｌｉｅｎａｒｄ方程从建模、实验到结果分析的全过程进行了可视化计算,并建立了一个对系统运动轨迹进行全面自动分析试验的可视化仿真框架,该文方法不仅不用传统程序代码对模型及算法编程,而且可方便地对系统进行多参数自动迭代试验及智能化分析。     

van der Pol系统极限环幅值的时滞控制

对含有时滞位移和时滞速度的van der Pol方程进行了研究,着重研究了时滞参数对van der Pol方程极限环幅值的影响．首先采用摄动法从理论上推导出极限环幅值与时滞参数之间的关系,分析时滞参数对幅值大小的影响,并着重讨论了不改变振动频率情况下对幅值的控制．最后用数值计算的方法验证了理论计算结果,结果表明数值计算结果与理论结果相当吻合．     

一类自治系统Hopf分叉及极限环幅值的时滞反馈控制

对含有非线性时滞位移的van der Pol—Duffing方程进行了研究,着重研究了时滞参数对van der Pol—Duffing系统Hopf分叉及极限环幅值的控制．首先采用摄动法从理论上推导出极限环幅值与时滞参数之间的关系,分析时滞参数对幅值大小的影响,并着重讨论了不改变振动频率情况下对幅值的控制．通过对零解的稳定性分析,得出Hopf分叉产生的条件．最后用数值计算的方法验证了理论计算结果,数值计算结果与理论结果相当吻合．     

混沌动力学系统的图形化建模及可视化仿真研究

该文研究非线性混沌动力学系统的图形化建模及可视化仿真计算问题．通过建立统一的可视化仿真实验框架,实现了一个高度一体化的混沌系统图形建模和可视化仿真环境;与基于常规高级程序设计语言的混沌计算机模拟技术相比,该方法避免了传统意义上的混沌方程算法编程和调试过程,亦可对混沌参数进行自动摄动分析及最优设计。     

混沌动力学系统的图形化建模可视化仿真研究

该文研究非线性混沌力学系统的图形化建模可视化仿真计算问题。通过建立统一的可视化仿真实验框架,实现了一具高度一体化的混沌系统图形建模和可视化仿真环境;与基于常规高级程序设计语言的混沌计算机模拟技术相比,该方法避免了传统意义上的混沌方程算法编程和调试过程,变可对混沌参数进行自动慑动分析及最优设计。     

强非线性系统的图形自动解算

通过构建统一的图形自动解算环境，对强非线性系统方程从建模，实验到结果分析的全过程进行了可视化解算。强调数据的图形处理，具有对复杂系统进行智能化多参数自动迭代运算的能力。     

基于深度学习的心律模拟算法及其应用

研究基于Van der Pol方程的心律模型有助于进行心律调控,为心律失常等心脏疾病治疗提供指导。针对已有Van der Pol方程时间尺度与真实生物心律不一致的问题,引入时间尺度参数,改进Van der Pol方程。针对心律模型的数值模拟问题,提出一种基于深度神经网络的算法。针对现有神经网络算法带来的假解问题,提出一种新的采样策略构建训练数据集。为进一步提高求解的精度,引入自适应采样策略。开展数值实验,将Runge-Kutta法给出的数值解与所提深度神经网络算法的计算结果进行对比。结果显示,深度神经网络算法的计算结果相对于Runge-Kutta数值解的最大平均误差不超过0.3%。还利用深度神经网络算法模拟牛蛙心脏搏动细胞动作电位信号以及外部驱动信号对该电位信号的调控作用,结果显示,深度神经网络算法可以很好地模拟动作电位的脉冲波形及外部驱动信号的调控作用。     

基于可视化计算的图形建模与参数寻优环境的设计与实现

研究了复杂非线性系统参数优化环境的可视化建模技术及软件实现问题。应用先进的仿真技术,采用面向对象和结构化设计方法,通过通用软件实现了复杂系统参数自动寻优环境的高度可视化的一体化;与基于常规高级和谐设计语言的参数寻优方法相比,该方法不仅不用传统程序代码对算法编程,而且可方便地对系统进行多参数自动迭代寻优试验及进行智能化分析。     

基于可视化计算的图形建模及参数寻优环境的设计与实现

研究了复杂非线性系统参数优化环境的可视化建模技术及软件实现问题．应用先进的仿真技术,采用面向对象和结构化设计方法,通过通用软件实现了复杂系统参数自动寻优环境的高度可视化的一体化;与基于常规高级和谐设计语言的参数寻优方法相比,该方法不仅不用传统程序代码对算法编程,而且可方便地对系统进行多参数自动迭代寻优试验及进行智能化分析．     

关于多尺度法的一个注记

以van der Pol系统为例,揭示了多尺度法在求解非线性系统三次以上近似解时似乎存在的二义性问题．通过将多尺度法的结果与KBM法所得结果进行比较,指出了如何利用多尺度法求解非线性系统三次以上近似解的正确步骤．     

Chaotic vibration of the one-dimensional linear wave equation with a van der Pol nonlinear boundary condition

The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic chaotic vibration by means of the total variation theory. Some results are derived on the exponential growth of total variation of the snapshots on the spatial interval in the long-time horizon when the map and the initial condition satisfy some conditions.     

随机Duffing-van der Pol系统响应的Chebyshev多项式逼近

对一类含随机参数的Dulling-van der Pol系统,运用Chebyshev多项式逼近法,将其转化成等价的确定性扩阶系统;通过求解等价系统在谐和激励下的稳态响应,可得Duffing-van der Pol系统相应的稳态随机响应,研究了当谐和激励的振幅变化时,含随机参数的Dulling-van der Pol系统的对称破裂分岔和倍周期分岔．数值模拟结果与数值解比较表明：正交多项式逼近法能有效地解决此类非线性随机动力系统的响应问题．     

时滞耦合van der Pol-Duffing振子环的动力学分析

研究含时滞的大规模van der Pol-Duffing耦合振子系统的非线性动力学.通过讨论特征方程根分布情况确定系统的稳定性,并在耦合时滞和强度平面上给出振幅死亡区域.结合数值算例,揭示同步和异步周期振荡、概周期运动以及混沌吸引子等现象.基于非线性振子电路和时滞电路,构建电路实验平台,有效验证理论和数值结果.研究结果表明,时滞可以显著影响系统动力学特性,如诱发振幅死亡、稳定性切换以及复杂振荡等.     

Damped oscillators: A continuous model for velocity dependent drag

We study a continuous model which describes the action of a velocity dependent drag on an oscillatory system. The solution of the resulting differential equation turns out to be asymptotic to a function which is expressible by means of the gamma and the hypergeometric functions. Among the applications, we consider that to the free oscillations around the fixed point which are ruled by van der Pol’s equation. A section is devoted to computational procedures and simulations.     

Global dynamics of two coupled parametrically excited van der Pol oscillators

Using a combination of analytical and numerical methods, the global bifurcations and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators are investigated in detail. With the aid of the method of multiple scales, the slow flow equations are obtained. Based on the slow flow equations, normal form theory and the techniques of choosing complementary space are applied to find the explicit expressions of the simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. By the simpler normal form, using the global perturbation method developed by Kovacic and Wiggins, the analysis of global bifurcation and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators is given. The results indicate that there exists a Silnikov type single-pulse homoclinic orbit for this class of system which implies the chaotic motions can occur. Numerical simulations are also given and verify the analytical predictions.     

On The Nonlinear Dynamics of Potential Relaxation Instability in a Weakly Magnetized Discharge Plasma

Some aspects of the nonlinear dynamics of the potential relaxation instability (PRI) that is triggered by a positive electrode (collector) in a weakly magnetized discharge plasma column are investigated experimentally. Some properties of the instability can be described by the van der Pol equation with an external force term. When the collector bias is increased the oscillations of plasma parameters become irregular. No low dimensional attractor can be positively identified for these signals. At high collector biases the plasma is probably in the state of a fully developed turbulence.     

Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems

In this article we investigate possible scenarios of pattern formations in reaction-diffusion systems with time fractional derivatives. Linear stability analysis is performed for different values of derivative orders. Results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations. Most attention is paid to two models: a fractional order reaction diffusion system with Bonhoeffer–van der Pol kinetics and to the Brusselator model.     

A reactive coordination scheme for a many-robot system

This paper presents a novel approach for coordinating a homogeneous system of mobile robots using implicit communication in the form of broadcasts. The broadcast-based coordination scheme was developed for the Army Ant swarm-a system of small, relatively inexpensive mobile robots that can accomplish complex tasks by cooperating as a team. The primary drawback, however, of the Army Ant system is that the absence of a central supervisor poses difficulty in the coordination and control of the agents. Our coordination scheme provides a global "group dynamic" that controls the actions of each robot using only local interactions. Coordination of the swarm is achieved with signals we call "heartbeats". Each agent broadcasts a unique heartbeat and responds to the collective behavior of all other heartbeats. We generate heartbeats with van der Pol oscillators. In this application, we use the known properties of coupled van der Pol oscillators to create predictable group behavior. Some of the properties and behaviors of coupled van der Pol oscillators are discussed in detail. We emphasize the use of this scheme to allow agents to simultaneously perform an action such as lifting, steering, or changing speed. The results of experiments performed on three actual heartbeat circuits are presented and the behavior of the realized system is compared to simulated results. We also demonstrate the application of the coordination scheme to global speed control.