采用组合参数的神经网络结构损伤检测分析研究

提出由结构前几阶固有频率变化率、频率变化比值和动柔度置信因子构成的组合参数作为神经网络的输入向量的方法进行结构损伤检测,全面分析了不同参数作为神经网络输入向量的损伤效果,利用数值模拟对悬臂梁、桁架结构进行分析,采用不同的输入参数进行比较·　分析结果表明,采用组合参数训练的神经网络,对结构损伤位置和程度识别较采用单一参数具有更好的识别效果·     

Finite dimensional Markov process approximation for stochastic time-delayed dynamical systems

This paper presents a method of finite dimensional Markov process (FDMP) approximation for stochastic dynamical systems with time delay. The FDMP method preserves the standard state space format of the system, and allows us to apply all the existing methods and theories for analysis and control of stochastic dynamical systems. The paper presents the theoretical framework for stochastic dynamical systems with time delay based on the FDMP method, including the FPK equation, backward Kolmogorov equation, and reliability formulation. A simple one-dimensional stochastic system is used to demonstrate the method and the theory. The work of this paper opens a door to various studies of stochastic dynamical systems with time delay.     

Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5

A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ≥ (2k + 1)2 - 1 for the perturbed Hamiltonian systems.     

Information and dynamical systems: a concrete measurement on sporadic dynamics

We present a method for the study of dynamical systems based on the notion of quantity of information. Measuring the quantity of information of a string by using data compression algorithms, it is possible to give a notion of orbit complexity of dynamical systems. In compact ergodic dynamical systems, entropy is almost everywhere equal to orbit complexity. We have introduced a new compression algorithm called CASToRe which allows a direct estimation of the information content of the orbits in the 0-entropy case. The method is applied to a sporadic dynamical system (Manneville map).     

不可压缩Navier-Stokes方程最优动力系统建模和分析

研究了同时满足任意速度边界条件和速度不可压条件的Navier-Stokes方程最优动力系统的建模方法.通过对方柱绕流问题的最优动力系统的建模与分析,发现该最优动力系统的动力学特性为极限环.同时,该最优动力系统仅使用了三个最优基函数就很好地描述了所有主要的流场特征和该问题的动力学特性,故满足任意速度边界条件和速度不可压条件Navier-Stokes方程最优动力系统的建模方法,能够用最少的基函数最大限度地描述复杂流体问题及其动力学特性.     

A note on kernel methods for multiscale systems with critical transitions

We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast‐slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near fast subsystem bifurcation points can be computed to leading order. This leading order approximation shows that the MMD depends intricately on the fast‐slow systems parameters, which can influence the detection of potential early‐warning signs before critical transitions. However, the MMD turns out to be an excellent binary classifier to detect the change‐point location induced by the critical transition. We cross‐validate our results by numerical simulations for a van der Pol‐type model.     

Conserved Quantities and Symmetries Related to Stochastic Dynamical Systems

The present article focuses on the three topics related to the notions of "conserved quantities" and "symmetries" in stochastic dynamical systems described by stochastic differential equations of Stratonovich type. The first topic is concerned with the relation between conserved quantities and symmetries in stochastic Hamilton dynamical systems, which is established in a way analogous to that in the deterministic Hamilton dynamical theory. In contrast with this, the second topic is devoted to investigate the procedures to derive conserved quantities from symmetries of stochastic dynamical systems without using either the Lagrangian or Hamiltonian structure. The results in these topics indicate that the notion of symmetries is useful for finding conserved quantities in various stochastic dynamical systems. As a further important application of symmetries, the third topic treats the similarity method to stochastic dynamical systems. That is, it is shown that the order of a stochastic system can be reduced, if the system admits symmetries. In each topic, some illustrative examples for stochastic dynamical systems and their conserved quantities and symmetries are given.     

Multi-particle dynamical systems and polynomials

Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.     

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED HYBRID DYNAMICAL SYSTEMS

In this paper, we consider a class of optimal control problem for the singularly perturbed hybrid dynamical systems. By means of variational method, we obtain the necessary conditions of the hybrid dynamical systems. Meanwhile, the existence of solution for the hybrid dynamical system is proved by the sewing method and the uniformly valid asymptotic expansion of the optimal trajectory is constructed by the boundary function method. Finally,an example is presented to illustrate the result.     

Monitoring change point for diffusion parameter based on discretely observed sample from stochastic differential equation models

Stochastic differential equation (SDE) models are useful in describing complex dynamical systems in science and engineering. In this study, we consider a monitoring procedure for an early detection of dispersion parameter change in SDE models. The proposed scheme provides a useful diagnostic analysis for phase I retrospective study and develops a flexible and effective control chart for phase II prospective monitoring. A standardized control chart is constructed, and a bootstrap method is used to estimate the mean and variance of the monitoring statistic. The control limit is obtained as an upper percentile of the maximum value of a standard Wiener process. The proposed procedure appears to have a manageable computational complexity for online implementation and also to be effective in detecting changes. We also investigate the performance of the exponentially weighted mean squared control charts for the continuous SDE processes. A simulation method is used to study the empirical sizes and the average run length characteristics of the proposed scheme, which also demonstrates the effectiveness of our method. Finally, we provide an empirical example for illustration. Copyright © 2014 John Wiley & Sons, Ltd.     

Output Stabilization for Nonminimum-Phase Systems Using Observer Backstepping

The article considers stabilization of the output programmed trajectory in nonminimum-phase dynamical systems of a special kind with unstable linear zero dynamics. An algorithm is proposed for the construction of a stabilizing control by observer backstepping. The backstepping method is generalized to nonminimum-phase nonlinear dynamical systems.     

Three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras

Dynamical systems attract much attention due to their wide applications. Many significant results have been obtained in this field from various points of view. The present paper is devoted to an algebraic method of integration of three-dimensional nonlinear time dependent dynamical systems admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras $L_4.$ The invariance of the relation between a dynamical system admitting nonlinear superposition and its Vessiot-Guldberg-Lie algebra is the core of the integration method. It allows to simplify the dynamical systems in question by reducing them to \textit{standard forms}. We reduce the three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras to 98 standard types and show that 86 of them are integrable by quadratures.     

Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5

A concrete numerical example of Z₆-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ⩾ (2k + I)² - 1 for the perturbed Hamiltonian systems.     

Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems

The notion of Lyapunov function plays a key role in the design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives. Furthermore, we present a method for automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is relatively complete in the sense that it is able to discover all polynomial RLFs with a given predefined template for any PDS. Therefore it can also generate all polynomial RLFs for the PDS by enumerating all polynomial templates.     

A class of nonstandard numerical methods for autonomous dynamical systems

A nonstandard finite difference method for solving autonomous dynamical systems is constructed. The proposed numerical method is computationally efficient and easy to implement. It is designed so that it preserves positivity of solutions and the local behavior of the dynamical system near equilibria.     

Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization

A novel method of estimation of the largest Lyapunov exponent for discrete maps is introduced and evaluated for chosen examples of maps described by difference equations or generated from non-smooth dynamical systems. The method exploits the phenomenon of full synchronization of two identical discrete maps when one of them is disturbed. The presented results show that this method can be successfully applied both for discrete dynamical systems described by known difference equations and for discrete maps reconstructed from actual time series. Applications of the method for mechanical systems with discontinuities and examples of classical maps are presented. The comparison between the results obtained by means of the known algorithms and novel method is discussed.     

One family of conformally Hamiltonian systems

We propose a method for constructing conformally Hamiltonian systems of dynamical equations whose invariant measure arises from the Hamiltonian equations of motion after a change of variables including a change of time. As an example, we consider the Chaplygin problem of the rolling ball and the Veselova system on the Lie algebra e*(3) and prove their complete equivalence.     

Generic bifurcations of low codimension of planar Filippov Systems

In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of this article is to develop a systematic method for studying local (and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequences.     

A method of continuous time approximation of delayed dynamical systems

This paper presents a new method to analyze response of linear and nonlinear dynamical systems with time delay. The method proposes a continuous time approximation of the delayed portion of the response. This leads to a high and finite dimensional state space formulation of the time-delayed system. The advantage of the current method lies in that the resulting finite dimensional state equations are in the standard state space form, making all the existing analysis methods and control design tools for linear and nonlinear dynamical systems amenable to the current approach. The method can also handle multiple independent time delays in a natural way. One- and two-dimensional dynamical systems with time delay are used to demonstrate the effectiveness of the method.     

A CSP Versus a Zonotope-Based Method for Solving Guard Set Intersection in Nonlinear Hybrid Reachability

Computing the reachable set of hybrid dynamical systems in a reliable and verified way is an important step when addressing verification or synthesis tasks. This issue is still challenging for uncertain nonlinear hybrid dynamical systems. We show in this paper how to combine a method for computing continuous transitions via interval Taylor methods and a method for computing the geometrical intersection of a flowpipe with guard sets, to build an interval method for reachability computation that can be used with truly nonlinear hybrid systems. Our method for flowpipe guard set intersection has two variants. The first one relies on interval constraint propagation for solving a constraint satisfaction problem and applies in the general case. The second one computes the intersection of a zonotope and a hyperplane and applies only when the guard sets are linear. The performance of our method is illustrated on examples involving typical hybrid systems.