Linear time-periodic dynamical systems: an H2 analysis and a model reduction framework

Linear time-periodic (LTP) dynamical systems frequently appear in the modelling of phenomena related to fluid dynamics, electronic circuits and structural mechanics via linearization centred around known periodic orbits of nonlinear models. Such LTP systems can reach orders that make repeated simulation or other necessary analysis prohibitive, motivating the need for model reduction. We develop here an algorithmic framework for constructing reduced models that retains the LTP structure of the original LTP system. Our approach generalizes optimal approaches that have been established previously for linear time-invariant (LTI) model reduction problems. We employ an extension of the usual H₂ Hardy space defined for the LTI setting to time-periodic systems and within this broader framework develop an a posteriori error bound expressible in terms of related LTI systems. Optimization of this bound motivates our algorithm. We illustrate the success of our method on three numerical examples.     

Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems and two linearly coupled Lorenz systems. Some sufficient conditions for synchronization are attained through rigorous mathematical theory. Compared with the results in the reference [Chaos, Solitons & Fractals 2002;14:529], the sufficient condition for the synchronization of two linearly coupled Lorenz systems is simpler and less conservative. Numerical simulations are provided for illustration and verification.     

On the stability of linear conservative gyroscopic systems

A simple criterion concerning the stability of linear gyroscopic conservative systems is developed. First, a sufficient condition for stability is established. The proof is based on a transformation converting the original autonomous system into a nonautonomous system, and applying Liapunov's direct method to the latter. Then a theorem is given which provides a necessary and sufficient condition for a restricted class of systems. Illustrative examples are provided.     

On the simultaneous diagonal stability of a pair of positive linear systems

In this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic Lyapunov function (CQLF) for a pair of positive linear time-invariant (LTI) systems.     

Stabilization for a class of second-order switched systems

This paper considers the asymptotic stabilization problem of second-order linear time-invariant (LTI) autonomous switched systems consisting of two subsystems with unstable focus equilibrium. More precisely, we find a necessary and sufficient condition for the origin to be asymptotically stable under the predesigned switching law. The result is obtained without looking for a common Lyapunov function or multiple Lyapunov function, but studying the locus in which the two subsystem's vector fields are parallel. Then the “most stabilizing” switching laws are designed which have translated the switched system into a piecewise linear system. Two numerical examples are presented to show the potential of the proposed techniques.     

Instabilities of Multihump Vector Solitons in Coupled Nonlinear Schrödinger Equations

Spectral stability of multihump vector solitons in the Hamiltonian system of coupled nonlinear Schrödinger (NLS) equations is investigated both analytically and numerically. Using the closure theorem for the negative index of the linearized Hamiltonian, we classify all possible bifurcations of unstable eigenvalues in the systems of coupled NLS equations with cubic and saturable nonlinearities. We also determine the eigenvalue spectrum numerically by the shooting method. In case of cubic nonlinearities, all multihump vector solitons in the nonintegrable model are found to be linearly unstable. In case of saturable nonlinearities, stable multihump vector solitons are found in certain parameter regions, and some errors in the literature are corrected.     

On analytical justification of phase synchronization in different chaotic systems

In analytical or numerical synchronizations studies of coupled chaotic systems the phase synchronizations have less considered in the leading literatures. This article is an attempt to find a sufficient analytical condition for stability of phase synchronization in some coupled chaotic systems. The method of nonlinear feedback function and the scheme of matrix measure have been used to justify this analytical stability, and tested numerically for the existence of the phase synchronization in some coupled chaotic systems.     

Multi-level substructuring combined with model order reduction methods

In many fields of engineering problems linear time-invariant dynamical systems (LTI systems) play an outstanding role. They result for instance from discretizations of the unsteady heat equation and they are also used in optimal control problems. Often the order of LTI systems is a limiting factor, since it becomes easily very large. As a consequence these systems cannot be treated efficiently without model reduction algorithms. In this paper a new approach for the combination of model order reduction methods and recent multi-level substructuring (MLS) techniques is presented. Similar multi-level substructuring methods have already been applied successfully to huge eigenvalue problems up to several millions of degrees of freedom. However, the presented approach does not make use of a modal analysis like former algorithms. Instead the original system is decomposed in smaller LTI systems which are treated with recent model reduction methods. Furthermore, the error which is induced by this substructuring approach is analysed and numerical examples based on the Oberwolfach benchmark collection are given in this paper.     

Linearly and nonlinearly bidirectionally coupled synchronization of hyperchaotic systems

To date, there have been many results about unidirectionally coupled synchronization of chaotic systems. However, much less work is reported on bidirectionally-coupled synchronization. In this paper, we investigate the synchronization of two bidirectionally coupled Chen hyperchaotic systems, which are coupled linearly and nonlinearly respectively. Firstly, linearly coupled synchronization of two hyperchaotic Chen systems is investigated, and a theorem on how to choose the coupling coefficients are developed to guarantee the global asymptotical synchronization of two coupled hyperchaotic systems. Analysis shows that the choice of the coupling coefficients relies on the bound of the chaotic system. Secondly, the nonlinearly coupled synchronization is studied; a sufficient condition for the locally asymptotical synchronization is derived, which is independent of the bound of the hyperchaotic system. Finally, numerical simulations are included to verify the effectiveness and feasibility of the developed theorems.     

STABILITIES OF （A,B,C）AND NPDIRK METHODS FOR SYSTEMS OF NEUTRAL DELAY—DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

Consider the following neutral delay-differential equations with multiple delays (NMDDE)$$y'(t)=Ly(t)+\sum_{j=1}^{m}[M_jy(t-\tau_j)+N_jy'(t-\tau_j)],\ \ t\geq 0, (0.1)$$ where $\tau>0$, $L, M_j$ and $N_j$ are constant complex- value $d×d$ matrices. A sufficient condition for the asymptotic stability of NMDDE system (0.1) is given. The stability of Butcher's (A,B,C)-method for systems of NMDDE is studied. In addition, we present a parallel diagonally-implicit iteration RK (PDIRK) methods (NPDIRK) for systems of NMDDE, which is easier to be implemented than fully implicit RK methos. We also investigate the stability of a special class of NPDIRK methods by analyzing their stability behaviors of the solutions of (0.1).     

Design of LMI‐based global sliding mode controller for uncertain nonlinear systems with application to Genesio's chaotic system

Design of a novel global sliding mode control law for the stabilization of uncertain nonlinear systems is presented in this article. A sufficient condition is derived using the Lyapunov theorem and linear matrix inequality to guarantee the asymptotical stability of the states and to improve the stability of the system. Under the uncertainty and nonlinearity effects, the reaching phase is eliminated and the chattering is reduced effectively and then, the robustness and performance of the system are improved. Lastly, the proposed method is applied on Genesio's chaotic system and the simulation results demonstrate the effectiveness of this technique. © 2014 Wiley Periodicals, Inc. Complexity 21: 94–98, 2015     

Simulation of the interaction of light with 3-D metallic nanostructures using a proper orthogonal decomposition-Galerkin reduced-order discontinuous Galerkin time-domain method

In this artice, we report on a reduced-order model (ROM) based on the proper orthogonal decomposition (POD) technique for the system of 3-D time-domain Maxwell's equations coupled to a Drude dispersion model, which is employed to describe the interaction of light with nanometer scale metallic structures. By using the singular value decomposition (SVD) method, the POD basis vectors are extracted offline from the snapshots produced by a high order discontinuous Galerkin time-domain (DGTD) solver. With a Galerkin projection and a second order leap-frog (LF₂) time discretization, a discrete ROM is constructed. The stability condition of the ROM is then analyzed. In particular, when the boundary is a perfect electric conductor condition, the global energy of the ROM is bounded, which is consistent with the characteristics of global energy in the DGTD method. It is shown that the ROM based on Galerkin projection can maintain the stability characteristics of the original high dimensional model. Numerical experiments are presented to verify the accuracy, demonstrate the capabilities of the POD-based ROM and assess its efficiency for 3-D nanophotonic problems.     

Global exponential stability for coupled retarded systems on networks: A graph-theoretic approach

In this paper, some coupled retarded systems on networks (CRSNs) are studied. Applying the idea of Razumikhin method and Mirchhoff’s matrix tree theorem, the sufficient conditions for global exponential stability of the CRSNs are obtained. Finally, an example of coupled retarded oscillator system on networks is given to illustrate the advantages of our results.     

Finite‐time stability of fractional‐order stochastic singular systems with time delay and white noise

In this article, the finite‐time stochastic stability of fractional‐order singular systems with time delay and white noise is investigated. First the existence and uniqueness of solution for the considered system is derived using the basic fractional calculus theory. Then based on the Gronwall's approach and stochastic analysis technique, the sufficient condition for the finite‐time stability criterion is developed. Finally, a numerical example is presented to verify the obtained theory. © 2016 Wiley Periodicals, Inc. Complexity 21: 370–379, 2016     

Razumikhin method to global exponential stability for coupled neutral stochastic delayed systems on networks

Global exponential stability for coupled neutral stochastic delayed systems on networks (CNSDSNs) is investigated in this paper. By means of combining the Razumikhin method with graph theory, some sufficient conditions that can be verified easily are derived to ensure the global exponential stability for CNSDSNs. Finally, a specific model of CNSDSNs is discussed, and numerical test manifests the effectiveness of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Stabilization of nonlinear systems in compound critical cases

In this paper, we study the stabilization of nonlinear systems in critical cases by using the center manifold reduction technique. Three degenerate cases are considered, wherein the linearized model of the system has two zero eigenvalues, one zero eigenvalue and a pair of nonzero pure imaginary eigenvalues, or two distinct pairs of nonzero pure imaginary eigenvalues; while the remaining eigenvalues are stable. Using a local nonlinear mapping (normal form reduction) and Liapunov stability criteria, one can obtain the stability conditions for the degenerate reduced models in terms of the original system dynamics. The stabilizing control laws, in linear and/or nonlinear feedback forms, are then designed for both linearly controllable and linearly uncontrollable cases. The normal form transformations obtained in this paper have been verified by using code MACSYMA.     

Heat transfer at high energy devices with prescribed cooling flow

We study the heat transfer from a high‐energy electric device into a surrounding cooling flow. We analyse several simplifications of the model to allow an easier numerical treatment. First, the flow variables velocity and pressure are assumed to be independent from the temperature which allows a reduction to Prandtl's boundary layer model and leads to a coupled nonlinear transmission problem for the temperature distribution. Second, a further simplification using a Kirchhoff transform leads to a coupled Laplace equation with nonlinear boundary conditions. We analyse existence and uniqueness of both the continuous and discrete systems. Finally, we provide some numerical results for a simple two‐dimensional model problem. Copyright © 2006 John Wiley & Sons, Ltd.     

对称耦合振子中同相解的稳定性

本文研究了一类耦合系统中同相解的稳定性,并给出了耦合Oregon振子中同相解 稳定性的充分条件.     

具有输入时滞的时滞关联不确定系统的鲁棒分散控制

研究了一类同时具有输入时滞以及不确定参数的时滞关联大系统的稳定性问题.基于所谓的还原法,给出一种新的状态反馈控制器的设计方法,这种方法的不同之处在于利用了时延的大小以及反馈控制的历史信息.根据Lyapunov稳定性理论得到了系统在控制器作用下稳定的充分条件,所有条件都化成可解的标准LMIs(Linear matrix inequalities)形式.文章最后给出了一个数值例子说明本文结果的可行性.