An ergodic classification of bidimensional linear systems

The ergodic structure of the projective flow induced by a family of bidimensional linear systems is studied. It is shown that the existence of a continuous invariant measure guarantees the existence of another measure, called linear by the authors, which provides substantial information upon the properties of the complex bundle. Some examples are given to illustrate the applicability of these results.     

A multiplicative ergodic theorem for rotation numbers

Given a vector fieldX on a Riemannian manifoldM of dimension at least 2 whose flow leaves a probability measure invariant, the multiplicative ergodic theorem tells us that-a.s. every tangent vector possesses a Lyapunov exponent (exponential growth rate) that is equal to one of finitely many basic exponents corresponding toX and. We prove that, in the case of a simple Lyapunov spectrum, every tangent plane-a.s. possesses a rotation number that is equal to one of finitely many basic rotation numbers corresponding toX and. Rotation in a plane is measured as the time average of the infinitesimal changes of the angle between a frame moved by the linearized flow and the same frame parallel-transported by a (canonical) connection.     

The Poincare-Bendixson theorem for monotone cyclic feedback systems

We prove the Poincare-Bendixson theorem for monotone cyclic feedback systems; that is, systems inR ⁿ of the form

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Topological Fixed Point Theorems Do Not Hold for Random Dynamical Systems

The aim of this paper is to demonstrate that topological fixed point theorems have no canonical generalization to the case of random dynamical systems. This is done by using tools from algebraic ergodic theory. We give a criterion for the existence of invariant probability measures for group valued cocycles. With that, examples of continuous random dynamical systems on a compact interval without random invariant points, which are an appropriate generalization of fixed points, are constructed.     

Affine Transformation in Random Iterated Function Systems

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LAGRANGE'S THEOREM FOR A CLASS OF NONHOLONOMICSYSTEMS AND ITS APPLICATION

I.IntroductionSinceE.T.Whittaker.proposedfoestabilit}'problellll'lofnonholononlicsystemsin1904forthefirsttime,thescholarsathomeandabroad11a\'emadealotofresearchesontheequilibriunlstabilityoflinearand11olllinearnonllolollolnicsystems,andhaveobtainedaseriesofimportantresultslZ--7].Hobbled'er,theexpositionandapplicationrelatedtoLagrange'stheorenlinthestabilityanalysisfornonholonomicsystemsisseldonlseenuptonow.Althoughitwasmentionedinreference[3].aspecialdiscussionhasnotbeencarriedoutyet.Asafam…     

Noether's theorem for nonholonomic systems of non-Chetaev's type with unilateral constraints in event space

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Necessary and sufficient stability conditions for invariant sets of nonlinear impulsive systems

Necessary and sufficient conditions for the uniform asymptotic stability of the invariant set of a nonlinear impulsive system are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 132–142, February 2008.     

Approximate sampling theorem for bivariate continuous function

An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation , a piece-wise linear function , and posseses an explicit computation formula . Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function .     

On the invariant theory of nonholonomic systems with constraints of non-Chetaev type

This paper deals with the theory of the differential invariant and integral invariant for a nonholonomic system with constraints of non-Chetaev type. It gives the restricted conditions of virtual displacement in velocity space for nonholonomic constraints of non-Chetaev type and extends the Jourdain principle and the canonical equation for the system. It presents and proves generalized Noether theorem, and gives generalized energy integral and cyclic integral for the system. Finally, the basic integral variants for the system are extended and the integral invariant of Poincaré-Cartan and the universal integral invariant of Poincaré for the system are obtained.     

A direct probabilistic approach to solve state equations for nonlinear systems under random excitation

In this paper, a direct probabilistic approach (DPA) is presented to formulate and solve moment equations for nonlinear systems excited by environmental loads that can be either a stationary or nonstationary random process. The proposed method has the advantage of obtaining the response’s moments directly from the initial conditions and statistical characteristics of the corresponding external exci-tations. First, the response’s moment equations are directly derived based on a DPA, which is completely independent of the It?/filtering approach since no specific assumptions regarding the correlation structure of excitation are made. By solving them under Gaussian closure, the response’s moments can be obtained. Subsequently, a multiscale algo-rithm for the numerical solution of moment equations is exploited to improve computational efficiency and avoid much wall-clock time. Finally, a comparison of the results with Monte Carlo (MC) simulation gives good agreement. Furthermore, the advantage of the multiscale algorithm in terms of efficiency is also demonstrated by an engineering example.     

Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.     

二阶随机参激系统的不变测度与Lyapunov指数

用统一的模型，研究了一类典型的二阶系统在宽带和窄带随机参数激励情形下，系统的不变测度与最大Ｌｙａｐｕｎｏｖ指数，由最大Ｌｙａｐｕｎｏｖ指数给出了系统几乎必然稳定的充分必要条件。     

Hausdorff Dimension of Invariant Sets for Random Dynamical Systems

Suppose X() is a compact random set, invariant with respect to a continuously differentiable random dynamical system (RDS) on a separable Hilbert space. It is shown that the Hausdorff dimension dim _H (X()) is an invariant random variable, and it is bounded by d, provided the RDS contracts d-dimensional volumes exponentially fast. Both exponential decrease of d-volumes as well as the approximation of the RDS by its linearization are assumed to hold uniformly in . The results are applied to reaction diffusion equations with additive noise and to two-dimensional Navier–Stokes equations with bounded real noise.     

A method toward the construction of Liapunov function for discrete time systems and determination of their stability

The purpose of this paper is to present a new method for constructing Liapunov function and determining the stability of discrete time systems with a computer on the basis of the similarity transformation theory by directly applying the system matrix of the system under discussion instead of solving the discrete Liapunov's matrix equation.     

A limit theorem for the solutions of slow-fast systems with fractional Brownian motion

A limit theorem which can simplify slow—fast dynamical systems driven by fractional Brownian motion with the Hurst parameter H inside the (1/2, 1) interval has been proved. The slow variables of the original system can be approximated by the solution of the simplified equations in the sense of mean square. An example is presented to illustrate the applications of the limit theorem.     

Energy principle of large deformation for living systems

Living system is a non-equilibrium open system having the function of self-organizationand self-control.The study of energy principle of living system involves two principal parts:mechanics and thermodynamics.The classical infinitesimal deformation theory andthermodynamics of equilibrium state are not sufficient to explain the complex motion ofliving system.We aim in this paper to describe the mechanical energy principle ofmacroscopic motion of living body based on large deformation non-symmetry stress fieldtheory.The principle of irreversible thermodynamics applied to living system will be left inanother paper.     

Finite-time stability with respect to a closed invariant set for a class of discontinuous systems

This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.