一类金融系统的分岔分析与混沌

将金融系统中的价格指数前的固定常数系数改进为弹性系数.利用Routh-Hurwitz定理和分岔定理,讨论了该弹性系数对改进后的金融系统的平衡点稳定性,Pitchfork分岔,Hopf分岔及混沌的影响.运用Matlab进行数值模拟,验证了所得到的理论结果.并给出参数变化时利率振幅的变化图和对应的相图,直观地展示了金融系统的稳定状态、周期状态及混沌状态的规律.     

一个经济周期模型的分岔与混沌

在Goodwin与Puu的宏观经济思想基础上,得到了一个推广的非线性动力学经济周期系统．首先用数值方法研究了此系统在特定参数条件下的全局分岔行为．然后结合最大Lyapunov指数,详细讨论了系统在分岔过程中动力学特征的转变．通过分析分岔图形发现在某些参数区间内倍周期分岔导致了混沌;在混沌区域内嵌有多个周期窗口;“加速数”值的增加可以促进经济的周期性运动．最后介绍了分岔和混沌分析得到的动力学性质对理解经济波动的应用．     

Host-Parasitoid系统的分岔与混沌控制

用数值模拟的方法,研究了Host-Parasitoid模型.该模型是一类非线性离散系统,反映了在一定的时间和空间内,寄生虫和寄宿主之间的生存状态.通过调节各种影响下的分岔参数,可以观察到系统具有周期泡,倍周期分叉,间歇混沌和Hopf分岔等复杂非线性动力学现象,揭示了系统通向混沌的途径.利用不同周期遍历下的奇怪吸引子和具有分形边界的吸引盆对系统的非线性特性进行了深入的探讨.最后利用参数开闭环控制法对系统的混沌状态进行了有效的控制.数值仿真和理论分析表明,选择相应的控制参数可将该系统的混沌状态控制到不同的稳定周期运动.     

Bifurcation and chaos analysis of atomic force microscope system

Atomic force microscope (AFM) is a very high-resolution type of scanning probe microscope, which is an essential characterization and actuation tool in modern nanoscience or engineering. This paper investigates the bifurcation and chaos behavior of the probe tip from AFM system by the differential transformation method (DTM). The dynamic behavior of the probe tip is characterized by reference to bifurcation diagrams, phase portraits, power spectra, Poincaré maps and maximum Lyapunov exponent plots produced using the time-series data obtained from DTM. The results indicate that the probe tip behavior is significantly dependent on the magnitude of the vibrational amplitude. Specifically, the probe tip motion changes from T-periodic to 3T-periodic, then from 2T-periodic to multi-periodic, and finally to chaotic motion with windows of periodic motion as the vibrational amplitude is increased from 0 to 2.0. Furthermore, it is demonstrated that the DTM is in good agreement for the considered system.     

机床无刷直流电机系统的分岔分析与控制

文章主要研究了机床无刷直流电机系统的Hopf分岔控制问题.首先,对系统进行分岔分析,通过计算极限环曲率系数判定系统的Hopf分岔类型;然后设计Washout滤波器对系统进行分岔控制,根据Hopf分岔理论给出使原系统Hopf分岔位置发生改变的参数条件,利用Normal Form方法计算出受控系统的Hopf分岔正规型,根据正规型的实部大小判定Hopf分岔类型,给出使原系统Hopf分岔类型发生改变的参数条件;并借助MATLAB软件对理论结果进行数值仿真,理论结果和数值仿真表明:控制器中的线性增益能使系统在所期望的参数值处发生Hopf分岔,甚至消除Hopf分岔,控制器中的非线性增益能改变原系统的Hopf分岔类型及极限环幅值的大小.研究结果对无刷直流电动机系统的工程实际具有一定的指导意义.     

On exponential stabilization of non-linear non-autonomous systems

Non-linear non-autonomous systems are considered. Conditions for the feedback exponential stabilizability are established.     

一种带绝对值项系统的分岔、激变与混沌

研究了一种含有绝对值项的三维微分动力系统,用李雅普诺夫方法得到了系统发生第一次Hopf分岔的条件.利用相轨迹图、分岔图、最大李雅普诺夫指数谱等非线性动力学分析方法,分析了该系统从规则运动转化到混沌运动的规律.该系统是按照Feigenbaum途径(倍周期分岔)通向混沌的,在混沌区域存在周期窗口.当参数达到激变临界点时,混沌吸引子和不稳周期轨道在吸引子边界上碰撞,发生边界激变,激变临界值的领域内还存在相对长时间的瞬态混沌过程.     

Stability analysis of non-autonomous linear systems by a matrix decomposition method

A matrix decomposition method is used to study the stability of the equilibrium state of non-autonomous linear systems. This technique systematically provides sufficient stability and asymptotic stability conditions for such systems, Several criteria regarding the stability of linear systems with time-varying coefficients are developed. A few examples illustrating the procedure are discussed. These examples include a system governed by the Mathieu equation, and a parametrically excited multidegree-of-freedom system.     

一类两自由度碰撞振动系统的Hopf分岔和混沌

分析了一类两自由度碰撞振动系统的周期运动,并通过计算Poincare映射的线性化矩阵,确定周期运动的稳定性.分析表明,在一定的参数条件下系统存在周期倍化分岔和Hopf分岔,并通过数值模拟方法得到了以Poincare截面上的不变圈表示的拟周期响应.简明地讨论了系统通向混沌的道路.     

随机及移动荷载激励下弹性梁分岔与混沌

移动荷载通过简支梁时,粗糙的梁表面会使移动荷载转变为随机激励.本文考虑梁的几何非线性因素,基于随机Melnikov理论确定了系统在均方意义下发生异宿分岔以及混沌的边界条件.利用数值随机Runge-Kutta方法对随机激励和周期激励共同作用下的系统响应进行了仿真计算,最大Lyapunov指数等数值结果描述了动力学行为变化过程.结果表明当荷载的速度一定时,梁跨中的非线性动力学行为受到质量和随机激励的共同影响,表面平整度较差的梁会增加混沌产生的可能性.     

非自治非线性DEDS的周时研究

对于一般的非自治系统，根据对偶定理给出一个通用的周时计算公式，并将其应用到几类非线性DEDS，得到了一些结果，其中包括关于分离系统特征值的01sder定理的一个简洁证明，一类具有特定结构的min—max系统在输入均匀序列情况下的周时公式，以及广义二分系统的周时限幅器等。这些结果对非线性DEDS的分析、综合及周时配置具有重要意义。     

Bifurcation and chaos of an airfoil with cubic nonlinearity in incompressible flow

Using a combination of analytical and numerical methods, the paper studies bifurcations and chaotic motions of a two-dimensional airfoil with cubic nonlinearity in incompressible flow. One type of critical points (characterized by a negative eigenvalue, a simple zero eigenvalue and a pair of purely imaginary eigenvalues) for the bifurcation response equations is considered. With the aid of the normal form theory, the explicit expressions of the critical bifurcation lines leading to incipient and secondary b...     

On optimal control of non-autonomous switched systems with a fixed mode sequence

We consider differentiability with respect to the switch times of the value function of an optimal control problem for a non-autonomous switched system. The control variables are the switch times between the modes and the input in each mode. We provide a method to compute the derivative of the cost function given a nominal input. Then, we view the optimal control problem as a parametrized optimization problem in which the switch times are the parameters and the optimization is over the set of feasible inputs of each mode. From this point of view, we provide conditions under which the continuity and differentiability of the optimal value function, that is the cost function optimized over the inputs, can be guaranteed.     

A generalized Bessel equation and synthesis of second-order non-autonomous systems

A parametric constraint for a general, second-order, linear, non-autonomous system to possess responses expressible in terms of the well-known Bessel functions is presented. In effect, this paper develops a process of synthesis (of systems with a Bessel typo of responses) which can be programmed using algebraic languages to obtain a practical tool of study.     

Finite time stability conditions for non-autonomous continuous systems

Lower and upper bounds for the quadratic cost functional of linear regulators are derived. These bounds are explicitly expressed in terms of some system parameters, arc independent of the initial conditions and are rather easy to compute. The results are based on a previous work by the authors concerning the lower bound. In this paper, the upper bound established is used to improve the lower bound, both being calculated by an iterative procedure to any degree of accuracy which is dependent on the number of iterations only.     

Preservation of exponential stability for linear non-autonomous functional differential systems

We consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation. As particular cases, the system includes models with concentrated delays and systems of integrodifferential equations. Our method is based on Bohl–Perron type theorems.     

Uniform semiglobal practical asymptotic stability for non-autonomous cascaded systems and applications

It is due to the modularity they provide that results for cascaded systems have proved their utility in numerous control applications as well as in the development of general control techniques based on “adding integrators”. Nevertheless, the standing assumptions in most of the present literature on cascaded systems is that, when decoupled, the subsystems constituting the cascade are uniformly globally asymptotically stable (UGAS). Hence existing results fail in the more general case when the subsystems are uniformly semiglobally practically asymptotically stable (USPAS). This situation is often encountered in control practice, e.g. in control of physical systems with external perturbations, measurement noise, unmodelled dynamics, etc. After giving a rigorous framework for the analysis of such stability properties, this paper generalizes previous results for cascades by establishing that, under a uniform boundedness condition on its solutions, the cascade of two USPAS systems remains USPAS. An analogous result is derived for uniformly semiglobally asymptotically stable (USAS) systems in cascade. Furthermore, we show the utility of our results in the PID control of mechanical systems affected by unknown non-dissipative forces and considering the dynamics of the DC motors.     

Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions

We consider approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. In this study, we use the theory of fractional powers and α-norms, so our results can be applied to systems where nonlinear terms include derivatives of spatial variables. We formulate and prove sufficient conditions for approximate controllability. We also give a sample application of our results.     

Identification and prediction of stochastic dynamical systems in a polynomial chaos basis

Non-parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation. In a class of such models, multi-dimensional Chebychev polynomials in the state variables are fitted to the observed dynamical state of the system. Due to the approximative nature of this non-parametric model as well as to various other sources of uncertainty such as measurement errors and non-anticipative excitations, the parameters of the model exhibit a scatter that is treated here in a probabilistic context. The statistics of these coefficients are related to the physical properties of the model being analyzed, and are used to endow the model predictions with a probabilistic structure. They are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy. The proposed methodology is quite simple and robust.