A theory for synchronization of dynamical systems

In this paper, a theory for synchronization of multiple dynamical systems under specific constraints is developed from a theory of discontinuous dynamical systems. The concepts on synchronization of two or more dynamical systems to specific constraints are presented. The synchronization, desynchronization and penetration of multiple dynamical systems to multiple specified constraints are discussed, and the necessary and sufficient conditions for such synchronicity are developed. The synchronicity of two dynamical systems to a single specific constraint and to multiple specific constraints is investigated. Finally, the synchronization and the corresponding complexity for multiple slave systems with multiple master systems are discussed briefly. The meaning of synchronization for dynamical systems with constraints is extended as a generalized, universal concept. The theory presented in this paper may be as a universal theory for dynamical systems. The paper provides a theoretic frame work in order to control slave systems which can be synchronized with master systems through specific constraints in a general sense.     

Hyperbolic systems of conservation laws with a symmetry

Hyperbolic systems of conservaiton laws with a symmetry are studied. Some peculiar phenomena for such systems are shown. Admissibility criteria for solutions to such systems are discussed. propagation and cancellation of initial oscillations for the systems are classified. As a byproduct of this study, an existence theorem of global solutions for the Cauchy of the systems is established.     

Sensitivity of non-autonomous discrete dynamical systems

In this paper, the sensitivity for non-autonomous discrete systems is investigated. First of all, two sufficient conditions of sensitivity for general non-autonomous dynamical systems are presented. At the same time, one stronger form of sensitivity, that is, cofinite sensitivity, is introduced for non-autonomous systems. Two sufficient conditions of cofinite sensitivity for general non-autonomous dynamical systems are presented. We generalized the result of sensitivity and strong sensitivity for autonomous discrete systems to general non-autonomous discrete systems, and the conditions in this paper are weaker than the correlated conditions of autonomous discrete systems.     

若干类时滞切换系统的稳定性问题

主要研究了镇定切换系统的鲁棒稳定性问题.用切换lyapunov函数方法,通过定义指标函数,讨论了基于切换lyapunov函数的若干类时滞切换系统的稳定性问题,用矩阵不等式研究了时滞,时滞摄动和不确定时滞的切换系统的鲁棒稳定性.     

Controllability and observability of coordinated linear systems

This paper develops concepts of controllability and observability for the class of coordinated linear systems, a special class of linear hierarchical systems with a top-to-bottom information structure. The usual concepts of reachability and indistinguishability for linear systems are refined, taking into account the decentralized nature of coordinated linear systems. The corresponding controllability and observability decompositions are derived, and several concepts of controllability and observability for coordinated linear systems are introduced and discussed. These results are then combined to a definition and characterization of stabilizability via dynamic measurement feedback for this class of systems.     

RECURSIVE SYSTEM IDENTIFICATION

Most of existing methods in system identification with possible exception of those for linear systems are off-line in nature, and hence are nonrecursive. This paper demonstrates the recent progress in recursive system identification. The recursive identification algorithms are presented not only for linear systems （multivariate ARMAX systems） but also for nonlinear systems such as the Hammerstein and Wiener systems, and the nonlinear ARX systems. The estimates generated by the algorithms are online updated and converge a.s. to the true values as time tends to infinity.     

On stability for switched linear positive systems

This paper addresses the stability properties of switched linear positive systems in continuous-time as well as in discrete-time. In the discrete-time case, some sufficient and necessary conditions for asymptotic stability are derived for pairs of second order systems. Similar conditions are also established for a finite number of second order systems. Furthermore, for higher order systems, some results on stability are provided in a similar manner. In particular, in this case, a common linear Lyapunov function guaranteeing the stability of the switched positive systems can be easily located by means of geometry properties. In the continuous-time case, a finite number of second order systems are considered. Some equivalent conditions for stability of such systems are developed.     

Stability analysis and stabilization of linear symmetric matrix-valued continuous,discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.     

微分代数系统的标准型和分支

我们首先把微分代数系统归结为受限微分方程,且讨论了受限微分系统的参数化问题,即系统的状态空间形式。然后,借助于受限系统的状态空间形式,我们得到了受限微分系统平行于常微分方程的线性化问题和Ｈｏｐｆ分支理论问题的有关结果。     

复杂系统的一般数学框架(Ⅰ)

复杂系统的基本和最简单的结构就是网络.根据这一思想,本系列论文拟发展一套处理复杂系统的新数学框架.本文详细论述了系统的概念、一般描述方法:系统=(硬部,软部,环境)和局整关系,包括子系统、元素与系统的关系和系统与系统的关系;给出了系统运算的基本法则;简要论述了系统的软、硬部之间的诱导转化.     

Stability and robustness of quasi-linear impulsive hybrid systems

Both hybrid dynamical systems and impulsive dynamical systems are studied extensively in the literature. However, impulsive hybrid systems are not yet well studied. Nonetheless, many physical systems exhibit both system switching and impulsive jump phenomena. This paper investigates stability and robust stability of a class of quasi-linear impulsive hybrid systems by using the methods of Lyapunov functions and Riccati inequalities. Sufficient conditions for stability and robust stability of those systems are established. Some examples are given to illustrate the applicability of our results.     

Locally polyhedral linear inequality systems

Linear systems of an arbitrary number of inequalities provide external representations for the closed convex sets in the Euclidean space. In particular, the locally polyhedral systems introduced in this paper are the natural linear representation for quasipolyhedral sets (those subsets of the Euclidean space whose nonempty intersections with polytopes are polytopes). For these systems the geometrical properties of the solution set are investigated, and their extreme points and edges are characterized. The class of locally polyhedral systems includes the quasipolyhedral systems, introduced by Marchi, Puente, and Vera de Serio in order to generalize the Weyl property of finite linear inequality systems.     

On the global existence and finite time blow-up of shadow systems

Shadow systems are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. In this paper, we study the global existence and blow-up phenomena for shadow systems. Our results show that even for these fundamental aspects, there are serious discrepancies between the dynamics of the reaction-diffusion systems and that of their corresponding shadow systems.     

Analysing periodicity,nonlinearity and transitional characteristics of nonlinear dynamic systems with Periodicity Ratio (PR)

The multiple-periodicity, nonlinearity and transitional characteristics of nonlinear dynamic systems subjected to external excitations are studied in this research. Diagnoses of the number and changing multiple-periodicities of Duffing’s systems are performed with implementation of the Periodicity Ratio (PR). The multiple-periodicity diagram is generated such that the periodicities and nonlinearity of the systems with respect to the system parameters can be graphically studied. The stability and convergence of the systems are investigated. The results of the research show that the number of period of periodicity of the systems increases continuously when certain system parameters increase. Transitional characteristics of the systems are also investigated. Both Lyapunov Exponents and Periodicity Ratio are implemented to diagnose the transitional routes of the systems. New symmetrical transition characters from periodicity to quasi-periodicity and chaos are displayed in terms of PR values. Comparing to Lyapunov Exponents, the Periodicity Ratio discloses more detailed and accurate transition information.     

Unifying theory for stability of continuous,discontinuous, and discrete-time dynamical systems

Continuous-time dynamical systems whose motions are continuous with respect to time (called continuous dynamical systems), may be viewed as special cases of continuous-time dynamical systems whose motions are not necessarily continuous with respect to time (called discontinuous dynamical systems, or DDS). We show that the classical Lyapunov stability results for continuous dynamical systems are embedded in the authors’ stability results for DDS (given in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474]), in the following sense: if the hypotheses for a given Lyapunov stability result for continuous dynamical systems are satisfied, then the hypotheses of the corresponding stability result for DDS are also satisfied. This shows that the stability results for DDS in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than was previously known, and that the quality of the DDS results therein is consistent with that of the classical Lyapunov stability results for continuous dynamical systems.By embedding discrete-time dynamical systems into a class of DDS with equivalent stability properties, we also show that when the hypotheses of the classical Lyapunov stability results for discrete-time dynamical systems are satisfied, then the hypotheses of the corresponding DDS stability results are also satisfied. This shows that the results for DDS in [H. Ye, A.N. Michel, L. Hou Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than previously known, having connections even with discrete-time dynamical systems!Finally, we demonstrate by the means of a specific example that the stability results for DDS are less conservative than corresponding classical Lyapunov stability results for continuous dynamical systems.     

NOTES ON GENERALIZED EXPONENTIAL DICHOTOMIES

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偏微分方程的吸引子

Abstract. In this paper the reaction-diffusion systems and the damped wave systems with non-linear terms of gradient form are studied ,and the conditions for making them to be gradient sys-tems are given. So the structure of attractors is clear and simple in some sense by the known re-sult. Some examples are given. Also the reaction-diffusion systems with general nonlinear termsare discussed.     

Majorants of the Dirichlet kernels and the Dini pointwise tests for generalized Haar systems

In this paper, we obtain five tests (three of which are symmetric) of pointwise convergence of Fourier series with respect to generalized Haar systems; the tests are similar to the Dini convergence tests. It is shown that the Dini convergence tests for Price systems are also valid for generalized Haar systems. It is also shown that the classicalDini convergence test does not apply, in general, even to generalized Haar systems, although the classical symmetric Dini test for generalized Haar systems is valid. Also upper bounds for the Dirichlet kernels for generalized Haar systems are obtained.     

On Stability of Symplectic Algorithms

1.FundamentalDeflnitionsLemma1.Thesolutionofalinearoofinarydtherentialequationwithcon8tantcoeffcientY=AYissta6leifalleigenvalue8ofAhaven0nP6sitivercalpartsandtheeigenvalueswithnullrealpartaresingleroots0ftheminimalp0lynomial.,/P\ThelinearHamiltoniansystemcanbeden0tedasZ=JSZwhereZ=(q),J=(ELs),andtheHamiltonianfuncti0nH(z)=ty.Lemma2.Thesolution80flinearHamiltoniansy8temsarecmticallysta6leifalleigenvaluesofJShavenullrsalpartandaresinglerootsojtheminitnalp0lyno?nial.Definiti0n1.Whenthemo…