Preliminary evaluation of electroencephalographic entrainment using thalamocortical modelling

Abstract: The concept of linked oscillators in biological control systems has long been established. Frequency entrainment is a predominant explanation behind many biological rhythms. In this paper a preliminary examination of electroencephalographic entrainment is made to survey the possibility and methods of achieving signal entrainment at the highest level of neurological organization and function. A model of the thalamocortical system is employed to generate simulated electroencephalographic signals and is tested in various configurations in the search for entrainment under very simple conditions. Additionally, an analysis of the coupled Van der Pol model of the circadian rhythm controller is performed to identify the possibility of affecting that system with a drastically different coupling input signal. We were able to conclude that overall signal shape can have a significant impact on the entrainment characteristics of the system. Due to the nature of the underlying mathematical structure of the model, by examining the circadian rhythm controller, we found that it is unsuitable for entrainment to an incident entraining signal of much higher frequency.     

Approximate solutions to Van der Pol damped nonlinear oscillators by means of He's energy balance method

In this article, He's energy balance method (EBM) is applied to solve Van der Pol damped nonlinear oscillators. Three examples of Van der Pol oscillators are presented and solved by this method to illustrate the effectiveness and convenience of the EBM. In this method, only one iteration leads to high accuracy of the solution. Comparisons are made between EBM results and exact solutions of the problems. The results show that the method can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other science.     

不确定Van der Pol混沌系统的模糊同步控制

本文研究了不确定Van der Pol混沌系统的同步问题,并进行了基于规则的模糊逻辑控制器(FLC)的控制。首先寻找主从Van der Pol混沌系统满足Lyapunov稳定性理论的条件,在此基础上建立模糊规则,设计模糊控制器,实现不确定混沌系统的同步。通过不确定VanderPol混沌系统的两组仿真结果,验证了模糊同步控制方法具有很好的鲁棒性。最后为了进一步验证该方法的有效性,本文在相同条件下,利用反馈控制的方法实现不确定主从VanderPol混沌系统的同步,然后再将此方法的仿真结果与本文的模糊同步控制方法的仿真结果在稳态误差及同步所需时间这两个方面进行对比分析。分析结果验证了本文同步方法的可行性及有效性。     

多自由度Van der Pol型系统振幅增大控制

研究多自由度Van der Pol型非线性振动系统振幅增大的控制,设计反馈控制器,用数值方法对控制系统的幅值进行了计算,绘制了在不同控制参数下,系统响应的时间历程曲线和极限环.研究表明通过调整控制参数,能够增大极限环的幅值,有工程应用价值,对高维系统的分岔控制研究有一定的理论意义.     

Van der Pol振动系统同步时间与反馈增益的关系

在受迫Van der Pol振动系统的近似解的基础上,获得驱动系统的虚拟轨线.将虚拟轨线代入驱动-响应振动系统的近似误差方程,再用多尺度法求得同步时间关于反馈增益的分析表达式,并且将数值与分析结果进行比较表明:用该方法求得的同步时间与反馈增益的关系和数值模拟结果相当一致.这方法也适用于研究自激Van der Pol振动系统.     

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.     

两个耦合Van der Pol振子系统的一阶近似守恒量

用直接积分法计算两个耦合Van der Pol振子系统的一阶近似守恒量,将两个耦合Van der Pol振子系统看成是未受微扰系统与微扰项的迭加,先通过坐标变换将未受微扰系统解耦,并对解耦系统的3种可能状态进行讨论,得到未受微扰系统的13个精确守恒量,再考虑微扰项对精确守恒量的影响,运用一阶近似守恒量的性质,得到1个稳定的一阶近似守恒量.另外,由13个精确守恒量直接得到13个平凡的一阶近似守恒量.     

Rejection of nonharmonic disturbances in a class of nonlinear systems with nonlinear exosystems

This paper proposes an asymptotic rejection algorithm for rejecting exotic disturbances in nonlinear systems. The disturbances, which are produced by nonlinear exosystems, are nonharmonic and periodic. A new internal model is proposed to deal with the disturbances. Further, an adaptive output feedback controller is designed to ensure that the system's state variables can asymptotically converge to zero, and the disturbances can be completely rejected. The proposed algorithm can be used in many applications, e.g. active vibration control, and the avoidance of nonharmonic distortion in nonlinear circuits. An example is performed to demonstrate that the proposed algorithm can completely reject the nonharmonic periodic disturbances produced by nonlinear exosystems. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A family of approximate inertial manifolds for a Van der Pol/FitzHugh–Nagumo perturbation problem

We investigate the stability of a family of approximate inertial manifolds (AIMs) obtained from an ODE containing a perturbation parameter. For two choices of the parameter, the dynamics associated with the equations are already well known: in one case, we have a Van der Pol equation, while in the other setting we obtain a FitzHugh–Nagumo equation. Recently, it has been shown that (a modified version of) each equation admits a sequence of AIMs which converges to the inertial manifold. We show that our model admits a family of AIMs depending on the perturbation parameter. We then investigate the stability of the family of AIMs as the perturbation parameter approaches two different vanishing coefficient limits. These results are intended to shed insight into the continuity properties of inertial manifolds.     

Bifurcation Control Of A Fractional‐Order Van Der Pol Oscillator Based On The State Feedback

In this paper, a dynamic state feedback is applied to control Hopf bifurcations arising from a fractional‐order Van Der Pol oscillator. The degree parameter indicating the strength of the nonlinear damping is chosen as the bifurcation parameter. It is shown that in the absences of the dynamic state feedback controller, the fractional‐order Van Der Pol oscillator loses the stability via the Hopf bifurcation early, and can maintain the stability only in a certain domain of the degree parameter. When applying the state feedback controller to the fractional‐order Van Der Pol oscillator, the onset of the undesirable Hopf bifurcation is postponed. Thus, the stability domain is extended, and the system possesses the stability in a larger parameter range. Numerical simulations are given to justify the validity of the dynamic state feedback controller in bifurcation controls.     

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.     

一类非线性系统参数辨识的新方法及其在生化发酵过程建模中的应用

本文运用按段多重Legendre多项式系(PMLP)[1],对一类参数可分离的非线性系统的参数辨识给出一种新方法,所提算法具有精度高、计算量小、不必具有待辨识参数的先验知识和可进行递推辨识等优点.该算法已成功地应用于只有少量实测数据的螺旋霉素菌体生长模型的动力学参数的辨识.结果表明,此算法是该类非线性系统的一种有效和简便的参数辨识方法.     

非线性控制系统线性化与极点配置的神经网络方法

提出一种同时实现非线性控制系统反馈线性化和极点配置的神经网络方法,并证明了在随机逼近意义下该神经网络控制系统的有效性.     

一类非线性系统参数辨识的新方法及其在生化发酵过程建模中的应用

本文运用按段多重Legendre多项式系(PMLP),对一类参数可分离的非线性系统的参数辨识给出一种新方法,所提算法具有精度高、计算量小、不必具有待辨识参数的先验知识和可进行递推辨识等优点。该算法已成功地应用于只有少量实测数据的螺旋霉素菌体生长模型的动力学参数的辨识。结果表明,此算法是该类非线性系统的一种有效和简便的参数辨识方法。     

半车非线性悬架的非平稳随机振动及控制

研究了1/2车非线性悬架模型在路面随机激励下的非平稳振动响应,并基于随机最优控制理论对其进行主动控制.首先利用等效线性化方法将具有非线性阻尼及迟滞刚度的非线性悬架模型线性化,然后将主动、被动悬架非平稳随机响应进行比较,结果表明非线性主动悬架的性能要优于被动悬架.最后,通过Monte-Carlo数值模拟验证了理论结果.     

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．     

On approximate linearization of nonlinear control systems

A method for the approximate linearization of nonlinear control systems based on the ‘state-space exact linearization’ method is presented. An explicit procedure, both for the single-input and for the multiple-input case, is given, which is straightforward to implement.     

非线性系统的分区域线性化控制及其在水轮机调节系统中的应用

本文初步解决了非线性系统的分区域线性化控制律的连续性和系统的稳定性问题,简要介绍了一个应用例子.