Dirac reduction for nonholonomic mechanical systems and semidirect products

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange–Dirac and Hamilton–Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler–Poincaré–Suslov equations with advected parameters and the implicit Lie–Poisson–Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin–Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.     

Determining functionals for random partial differential equations

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional random dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results.We apply our results to the 2D Navier-Stokes equations forced by a white noise.     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations

The reaction-diffusion equations are approximated by a fully discrete system: a Legendre-Galerkin approximation for the space variables and a semi-implicit scheme for the time integration. The stability and the convergence of the fully discrete system are established. It is also shown that, under a restriction on the space dimension and the growth rate of the nonlinear term, the approximate attractors of the discrete finite dimensional dynamical systems converge to the attractor of the original infinite dimensional dynamical systems. An error estimate of optimal order is derived as well without any further regularity assumption.     

Control by Interconnection and Energy Shaping of the Timoshenko Beam

In this paper, the dynamical control of a mixed finite and infinite dimensional mechanical system is approached within the framework of port Hamiltonian systems. In particular, a flexible beam, modeled according to the Timoshenko theory and in distributed port Hamiltonian form, with a mass under gravity field connected at a free end, is considered. The control problem is approached by generalization of the concept of structural invariant (Casimir function) to the infinite dimensional case and the so-called control by interconnection technique is extended to the infinite dimensional case. In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e., by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipation effect is introduced.     

离散的Burgers-Ginzburg-Landau方程组的吸引子

文章通过对空间变量的有限差分方法离散了具有周期边值的Ｂｕｒｇｅｒｓ Ｇｉｎｚｂｕｒｇ Ｌａｎｄａｕ方程组．研究了这个离散方程组初值问题解的适定性．证明了当差分网格足够大时离散方程组存在吸引子，并得到了吸引子的Ｈａｕｓｄｏｒｆｆ维数和分形维数的上界估计．这个上界不会随着网格的加细而无限增大，因此数值分析离散的有限维系统的吸引子可以近似探讨原无限维系统的吸引子．     

广义同步化流形的Holder连续性

证明了两个不同的混沌系统线性耦合时能实现广义同步化,在一定条件下广义同步化流形是Holder连续的.采用的思想是Temam的无穷维动力系统的惯性流形理论的改进.在线性耦合下两个混沌系统具有吸收集和吸引子的基础上,通过定义在一个函数类上的映射满足Schauder不动点定理,从而得到广义同步化流形,该广义同步化流形具有不变性.又证明了存在分数维的指数吸引子,指数吸引子与广义同步流形的交集具有指数吸引性.数值仿真证实了理论的正确性.在驱动系统和响应系统外引入辅助系统,辅助系统与响应系统的完全同步化表明了驱动系统和响应系统的广义同步化.     

函数空间多体挠性结构系统动力学、稳定性与控制的研究

利用现代数学方法,在函数空间中研究了一类无穷维系统动力学、稳定性与控制问题.首先提出并建立了具有阻尼、陀螺部件和约束阻尼的多拓扑结构多挠体分布参数系统动力学控制模型;其次给出并论证了多体挠性结构特征、系统分析结果--可控可观性充要条件、稳定性理论和系统的渐近性质.研究的结果扩充和发展了本领域关于多挠体系统动力学与控制的理论成果,具有重要的工程意义.     

Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field. Oblatum: 6-XI-1998 & 12-VI-1998 / Published online: 14 January 1999     

Pathological asymptotic behavior of control systems in Banach space

New phenomena arising when a linear dynamical system is defined on an infinite dimensional Banach space, although negligible from an engineering standpoint when only a finite time-interval is considered, become crucial when the asymptotic (feedback) behavior of the system is of interest. Pathologies with respect to the correspondent finite dimensional case are displayed even when the operator acting on the state is bounded.In particular, although in such case, the classical controllability and observability theory admits a natural generalization to infinite dimensions, the finite dimensional relationships between controllability and stabilizability fails. A few examples are given of systems that are approximately controllable and yet are not stabilizable: Moreover, such examples are drawn from a class of systems that can never be exactly controllable. The analysis is carried out using the perturbation theory of the spectrum. Another new feature of the infinite dimensionality of the state space is that even if the spectrum of an operator has the max of its real part equal to 0, yet the associated homogeneous differential equation may be globally asymptotically stable: Its consequence on stabilizability is also examined.     

一类非线性抛物型变分不等方程的全局吸引子及弱近似惯性流形

对由一类非线性抛物型变分不等方程所描述的无穷维动力系统，给出了存在全局吸引子及弱近似惯性流形的充分条件．     

Entropy,Chaos, and Weak Horseshoe for Infinite‐Dimensional Random Dynamical Systems

In this paper, we study the complicated dynamics of infinite‐dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we prove if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li‐Yorke. The complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system.© 2017 Wiley Periodicals, Inc.     

Large dynamics of Yang–Mills theory: mean dimension formula

We study the Yang–Mills anti-self-dual (ASD) equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover becomes an infinite dimensional and infinite entropy system. We study the mean dimension of this huge dynamical system. Mean dimension is a topological invariant of dynamical systems introduced by Gromov. We prove the exact formula of the mean dimension by developing a new technique based on the metric mean dimension theory of Lindenstrauss–Weiss.     

From phenomena of synchronization to exact synchronization and approximate synchronization for hyperbolic systems

In this survey paper, the synchronization will be initially studied for infinite dimensional dynamical systems of partial differential equations instead of finite dimensional systems of ordinary differential equations,and will be connected with the control theory via boundary controls in a finite time interval. More precisely,various kinds of exact boundary synchronization and approximate boundary synchronization will be introduced and realized by means of fewer boundary controls for a coupled system of wave equations with Dirichlet boundary controls. Moreover, as necessary conditions for various kinds of approximate boundary synchronization, criteria of Kalman's type are obtained. Finally, some prospects will be given.     

Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

Long-time convergence of numerical approximations for 2D GBBM equation

We study the long-time behavior of the finite difference solution to the generalized BBM equation in two space dimensions with dirichlet boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems. Numerical experiment results show that the theory is accurate and the schemes are efficient and reliable.     

On synchronization of a forced delay dynamical system via the Galerkin approximation

A forced scalar delay dynamical system is analyzed from the perspective of bifurcation and synchronization. In general first order differential equations do not exhibit chaos, but introduction of a delay feedback makes the system infinite dimensional and shows chaoticity. In order to study the dynamics of such a system, Galerkin projection technique is used to obtain a finite dimensional set of ordinary differential equations from the delay differential equation. We compare the results of simulation with those obtained from direct numerical simulation of the delay equation to ascertain the accuracy of the truncation process in the Galerkin approximation. We have considered two cases, one with five and the other with eight shape functions. Next we study two types of synchronization by considering coupling of the above derived equations with a forced dynamical system without delay. Our analysis shows that it is possible to have synchronization between two such systems. It has been shown that the chaotic system with delay feedback can drive the system without delay to achieve synchronization and the opposite case is also equally valid. This is confirmed by the evaluation of the conditional Lyapunov exponents of the systems.     

A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems

The Liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations. Extensions to infinite dimensional settings have been studied by several authors (see e.g. Haraux, 1991 [11], and Komornik and Zuazua, 1990 [17] and references therein). Results on optimal energy decay rates under general conditions of the feedback is far from being complete. The purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation. We present a general approach based on convexity arguments to establish sharp optimal or quasi-optimal upper energy decay rates for these systems, and on comparison principles based on the dissipation property, and interpolation inequalities (in the infinite dimensional case) for lower bounds of the energy. We stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi-discretized nonlinear damped vibrating PDE's. A part of this approach has been introduced in Alabau-Boussouira (2004, 2005) [1] and [2]. In the present paper, we identify a new, simple and explicit criteria to select a class of nonlinear feedbacks, for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in Alabau-Boussouira (2004, 2005) [1] and [2]. Moreover, we prove optimality of the decay rates for this class, in the finite dimensional case. This class includes a wide range of feedbacks, ranging from very weak nonlinear dissipation (exponentially decaying in a neighborhood of zero), to polynomial, or polynomial-logarithmic decaying feedbacks at the origin. In the infinite dimensional case, we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation. This principle relies on suitable interpolation inequalities. It allows us to give lower bounds for the energy of smooth initial data for the one-dimensional wave equation with a distributed polynomial damping, which improves Haraux (1995) [12] lower estimate of the energy for this case. We also establish lower bounds in the multi-dimensional case for sufficiently smooth solutions when such solutions exist. We further mention applications of these various results to several classes of PDE's, namely: the locally and boundary damped multi-dimensional wave equation, the locally damped plate equation and the globally damped coupled Timoshenko beams system but it applies to several other examples. Furthermore, we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems. We illustrate these results on the one-dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples. These optimal rates are not uniform with respect to the discretization parameter. We also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin.     

The convective Cahn–Hilliard equation

We consider the convective Cahn–Hilliard equation with periodic boundary conditions as an infinite dimensional dynamical system and establish the existence of a compact attractor and a finite dimensional inertial manifold that contains it. Moreover, Gevrey regularity of solutions on the attractor is established and used to prove that four nodes are determining for each solution on the attractor.     

(Non-)expansivity in functional envelopes

We study expansivity in functional envelopes of dynamical systems. Our main result is that if the phase space of the original system contains an arc or if it contains a free infinite zero dimensional set, then the expansivity in the functional envelope with Hausdorff metric is impossible. This is in contrast with the fact that, when considering the uniform metric in functional envelopes, the expansivity of a system is equivalent with the expansivity of its functional envelope.