Nesting inertial manifolds for reaction and diffusion equations with large diffusivity

We study the asymptotic behaviour in large diffusivity of inertial manifolds governing the long time dynamics of a semilinear evolution system of reaction and diffusion equations. A priori, we review both local and global dynamics of the system in scales of Banach spaces of Hilbert type and we prove the existence of a universal compact attractor for the equations. Extensions yield the existence of a family of nesting inertial manifolds dependent on the diffusion of the system of equations. It is introduced an upper semicontinuity notion in large diffusivity for inertial manifolds. The limit inertial manifold whose dimension is strictly less than those of the infinite dimensional system of semilinear evolution equations is obtained.     

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.     

Orbital stability of standing waves for semilinear wave equations with indefinite energy

The orbital stability of standing waves for semilinear wave equations is studied in the case that the energy is indefinite and the underlying space domain is bounded or a compact manifold or whole Rn with n?2. The stability is determined by the convexity on ω of the lowest energy d(ω) of standing waves with frequency ω. The arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows.     

Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs

We obtain global and local theorems on the existence of invariant manifolds for perturbations of nonautonomous linear differential equations assuming a very general form of dichotomic behavior for the linear equation. Besides some new situations that are far from the hyperbolic setting, our results include, and sometimes improve, some known stable manifold theorems.     

Center manifolds for periodic functional differential equations of mixed type

We study the behaviour of solutions to nonlinear functional differential equations of mixed type (MFDEs), that remain sufficiently close to a prescribed periodic solution. Under a discreteness condition on the Floquet spectrum, we show that all such solutions can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. This generalizes the results that were obtained previously in [H.J. Hupkes, S.M. Verduyn Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Differential Equations 19 (2007) 497-560] for bifurcations around equilibrium solutions to MFDEs.     

The fundamental existence theorem on invariant fiber bundles

For discrete dynamical systems the theory of invariant manifolds is well known to be of vital importance. In terms of difference equations this theory is basically concerned with autonomous equations. However, the crucial and currently most difficult questions in this field are related to non-periodic, in particular chaotic motions. Since this topic - even in the autonomous context is an intrinsically time-variant matter. There is and urgent need for a non-autonomous version of invariant manifold theory. In this paper we present we present a very general version of the classical result on stable and unstable manifolds for hyperbolic fixed points of diffeomorphisms. In fact, we drop the assumption of invertibility of the mapping, we consider non-autonomous difference equations rather than mappings In effect, we generalize the notion of invariant manifold to the concept of invariant fiber bundle.     

Invariant manifolds for singularly perturbed linear functional differential equations

The existence of “slow” and “fast” manifolds, and of invariant manifolds approaching the manifold of orbits of the degenerate system, is discussed for singularly perturbed systems of linear retarded functional differential equations (FDE). It is shown that these manifolds exist only in very degenerate situations and, consequently, the geometry of the flow of singularly perturbed ordinary differential equations does not generalize to FDEs.     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

An Intrinsic Geometric Formulation of the Equilibrium Equations in Continuum Mechanics

The theory of invariant continuum mechanics is based on the concept that forces and stresses are defined as elements of the cotangent bundle of the configuration manifold. While body and physical space are modeled as differentiable manifolds, the infinite dimensional configuration manifold is given by all configurations of the body in the physical space. In this paper a virtual work principle is postulated which leads together with an induced traction stress and Stokes' theorem directly to the local equilibrium equations and the traction boundary conditions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Center manifolds of infinite dimensions I: Main results and applications

In this paper, the first of a bipartite work, we consider an abstract, nonautonomous system of evolution equations of hyperbolic type, related to semilinear wave equations. Theorem 1 states that under certain assumptions the system admits a global center manifold, or equivalently a global decoupling function which is continuously differentiable with respect to its arguments, among which timet occurs. The difficult proof is presented in part II, i.e. the continuation of the present paper. For purposes of applications a local version of Theorem 1 is proved, i.e. the local center manifold Theorem 2. We obtain a series of applications both to abstract, nonautonomous wave equations and to concrete nonautonomous, semilinear wave equations subject to Neumann and Dirichlet boundary conditions.     

Surgery and the Yamabe Invariant

We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular: for a compact connected manifold M with no metric of positive scalar curvature, we prove that the Yamabe invariant of any manifold obtained by performing surgery on spheres of codimension greater than 2 on M is not smaller than the invariant of M. Submitted: August 1998.     

Discrete inertial manifolds

This work is devoted to attractive invariant manifolds for nonautonomous difference equations, occurring in the discretization theory for evolution equations. Such invariant sets provide a discrete counterpart to inertial manifolds of dissipative FDEs and evolutionary PDEs. We discuss their essential properties, like smoothness, the existence of an asymptotic phase, normal hyperbolicity and attractivity in a nonautonomous framework of pullback attraction. As application we show that inertial manifolds of the Allen–Cahn and complex Ginzburg–Landau equation persist under discretization. For the Ginzburg–Landau equation we can also estimate the dimension of the inertial manifold. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

周期流形的不变环面和次调和分支

本文通过精华Ｆｌｏｑｕｅｔ方法在周期流形的周期轨道邻域建立起适当的局部坐标，然后应用平均法和积分流形及Ｆｅｎｉｃｈｅｌ不变流形理论来证明不变环面和次调和轨道的存在性和法向双曲性．大多数传统的假设被放弃，而大多数已知的结果被推广．文中还给出了一个例子作为应用     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

The liouville property and boundary value problems for semilinear elliptic equations on noncompact Riemannian manifolds

We address the asymptotic behavior of solutions to semilinear equations on noncompact Riemannian manifolds. Under study is the relation between the solvability of some boundary and exterior boundary value problems as well as conditions for the fulfillment and stability of Liouville-type theorems for the solutions to semilinear equations on these manifolds.     

Numerical Taylor expansions of invariant manifolds in large dynamical systems

Summary. In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear (black box) solver and a low dimensional invariant subspace is available, but for which methods like the QR–Algorithm are considered to be too expensive. Our method is based on an analysis of the multilinear Sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions. These conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold. The final algorithm requires the solution of several large linear systems with a bordered Jacobian. To these systems we apply a block elimination method recently developed by Govaerts and Pryce [12, 14]. Received March 12, 1996 / Revised version reveiced August 8, 1997     

伴随奇点分支的退化异宿轨道的保存和横截性

本文应用指数三分性和不变流形的局部几何表示方法,给出异宿流形上的轨道当两个奇点经历超;临界分支和摄动时保存和横截的条件．     

Classification of homotopy Wall's manifolds

Complete PL and topological classification and partial smooth classification of manifolds homotopy equivalent to a Wall's manifold (defined as a mapping torus of a Dold manifold), introduced by Wall in his 1960 Annals paper on cobordism, have been done by determining: (1) the normal invariants of Wall's manifolds, (2) the surgery obstruction of a normal invariant and (3) the action of the Wall surgery obstruction groups on the smooth, PL and homeomorphism classes of homotopy Wall's manifolds (to be made precise in the body of the paper). Consequently classification results of automorphisms (self homeomorphisms, and self PL-homeomorphisms) of Dold manifolds follow.     

Slow and Fast Variables in Non-autonomous Difference Equations

In this paper we present an existence and smoothness result for center-like invariant manifolds of non-autonomous difference equations with slow and fast state-space variables. This result can be seen as a first step to obtain Fenichel's geometric theory for difference equations. Hereby, our basic tool is an abstract integral manifold theorem.     

Persistence of Invariant Manifolds for Nonlinear PDEs

We prove that under certain stability and smoothing properties of the semi-groups generated by the partial differential equations that we consider, manifolds left invariant by these flows persist under C ¹ perturbation. In particular, we extend well-known finite-dimensional results to the setting of an infinite-dimensional Hilbert manifold with a semi-group that leaves a submanifold invariant. We then study the persistence of global unstable manifolds of hyperbolic fixed points, and as an application consider the two-dimensional Navier–Stokes equation under a fully discrete approximation.Finally, we apply our theory to the persistence of inertial manifolds for those PDEs that possess them.