Nested sequences of index filtrations and continuation of the connection matrix

AbstractIn this paper, we prove the existence of nested sequences of index filtrations for convergent sequences of (admissible) semiflows on a metric space. This result is new even in the context of flows on a locally compact space. The nested index filtration theorem implies the continuation of homology index braids which, in turn, implies the continuation of connection matrices in the infinite-dimensional Conley index theory

The Conley index and chaos

Ph.D.Konstantin Mischaiko

The suspension isomorphism for homology index braids

Let $X$ be a metric space, $\pi$ be a local semiflow on $X$, $k\in\mathbb N$, $E$ be a $k$-dimensional normed space and $\widetilde\pi$ be the semiflow generated by the equation $\dot y=Ly$, where L\co E\to E is a linear map whose all eigenvalues have positive real parts. We show in this paper that for every admissible isolated $\pi$-invariant set $S$ there is a well-defined isomorphism of degree $-k$ from the homology categorial Conley-Morse index of $(\pi\times\widetilde\pi,S\times\{0\})$ to the homology categorial Conley-Morse index of $(\pi,S)$ such that the family of these isomorphisms commutes with homology index sequences. In particular, given a partially ordered Morse decomposition $(M_i)_{i\in P}$ of $S$ there is an isomorphism of degree $-k$ from the homology index braid of $(M_i\times\{0\})_{i\in P}$ to the homology index braid of $(M_i)_{i\in P}$, so $C$-connection matrices of $(M_i\times\{0\})_{i\in P}$ are just $C$-connection matrices of $(M_i)_{i\in P}$ shifted by $k$ to the right

Morse decompositions in the absence of uniqueness

In this paper we define attractors and Morse decompositions in an abstract framework of curves in a metric space. We establish some basic properties of these concepts including their stability under perturbations. This extends results known for flows and semiflows on metric spaces to large classes of ordinary or partial differential equations with possibly nonunique solutions of the Cauchy problem. As an application, we first prove a Morse equation in the context of a Conley index theory which was recently defined in [M. Izydorek and K. P. Rybakowski, On the Conley index in Hilbert spaces in the absence of uniqueness , Fund. Math.] for problems without uniqueness, and then apply this equation to give an elementary proof of two multiplicity results for strongly indefinite elliptic systems previously obtained in [S. Angenent and R. van der Vorst, A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology , Math. Z. 231 (1999), 203–248] using Morse-Floer homology

On the suspension isomorphism for index braids in a singular perturbation problem

We consider the singularly perturbed system of ordinary differential equations \aligned \varepsilon\dot y&=f(y,x,\varepsilon), \\ \dot x&=h(y,x,\varepsilon) \endaligned \leqno(E_\varepsilon) on Y\times \Cal{M}, where $Y$ is a finite dimensional normed space and \Cal{M} is a smooth manifold. We assume that there is a reduced manifold of $(E_\varepsilon)$ given by the graph of a function \phi\co \Cal{M}\to Y and satisfying an appropriate hyperbolicity assumption with unstable dimension $k\in{\mathbb N}_0$. We prove that every Morse decomposition $(M_p)_{p\in P}$ of a compact isolated invariant set $S_0$ of the reduced equation $$\dot x=h(\phi(x),x,0)$$ gives rises, for $\varepsilon> 0$ small, to a Morse decomposition $(M_{p,\varepsilon})_{p\in P}$ of an isolated invariant set $S_\varepsilon$ of $(E_\varepsilon)$ such that $(S_\varepsilon,(M_{p,\varepsilon})_{p\in P})$ is close to $(\{0\}\times S_0,(\{0\}\times M_p)_{p\in P})$ and the (co)homology index braid of $(S_\varepsilon,(M_{p,\varepsilon})_{p\in P})$ is isomorphic to the (co)homology index braid of $(S_0,(M_{p})_{p\in P})$ shifted by $k$ to the left

Morse decompositions in the absence of uniqueness, II

This paper is a sequel to our previous work [ Morse decompositions in the absence of uniqueness , Topol. Methods Nonlinear Anal. 18 (2001), 205–242]. We first extend the concept of $\mathcal{T}$-Morse decompositions to the partially ordered case and prove a generalization of a result by Franzosa and Mischaikow characterizing partially ordered $\mathcal{T}$-Morse decompositions by the so-called $\mathcal{T}$-attractor semifiltrations. Then we extend the (regular) continuation result for Morse decompositions from [ Morse decompositions in the absence of uniqueness , Topol. Methods Nonlinear Anal. 18 (2001), 205–242] to the partially ordered case. We also define singular convergence of families of ``solution'' sets in the spirit of our previous paper [ On a general Conley index continuation principle for singular perturbation problems , Ergodic Theory Dynam. Systems 22 (2002), 729–755] and prove various singular continuation results for attractor-repeller pairs and Morse decompositions. We give a few applications of our results, e.g. to thin domain problems. The results of this paper are a main ingredient in the proof of regular and singular continuation results for the homology braid and the connection matrix in infinite dimensional Conley index theory. These topics are considered in the forthcoming publications [ Continuation of the connection matrix in infinite-dimensional Conley index theory ] and [ Continuation of the connection matrix in singular perturbation problems ]

On convergence and compactness in parabolic problems with globally large diffusion and nonlinear boundary conditions

We establish some abstract convergence and compactness results for families of singularly perturbed semilinear parabolic equations and apply them to reaction-diffusion equations with nonlinear boundary conditions and large diffusion. This refines some previous results of [R. Willie, A semilinear reaction-diffusion system of equations and large diffusion , J. Dynam. Differential Equations 16 (2004), 35-63]

Resolvent convergence for Laplace operators on unbounded curved squeezed domains

We establish a resolvent convergence result for the Laplace operator on certain classes of unbounded curved squeezed domains \Omega_\eps as \eps\to0. As a consequence, we obtain Trotter-Kato-type convergence results for the corresponding family of $C^0$-semigroups. This extends previous results obtained by Antoci and Prizzi in \cite{\rfa{AP}} in the flat squeezing case

Conley Index Continuation and Thin Domain Problems

Given > 0 and a bounded Lipschitz domain in R M R N let := f (x; y) j (x; y) 2 g be the -squeezed domain. Consider the reaction-diusion equation ( ~ E ) u t = u + f(u) on with Neumann boundary condition. Here f is an appropriate nonlinearity such that ( ~ E ) generates a (local) semiow ~ on H 1( ). It was proved by Prizzi and Rybakowski (J. Di. Equations, to appear), generalizing some previous results of Hale and Raugel, that there are a closed subspace H 1 s( of H 1( 1 a closed subspace L 2 s( of L 2( and a sectorial operator A0 on L 2 s( such that the semiow 0 dened on H 1 s( by the abstract equation _ u +A0u = ^ f(u) is the limit of the semiows ~ as ! 0 + . In this paper we prove a singular Conley index continuation principle stating that every isolated invariant set K0 of 0 can be continued to a nearby family ~ K of isolated invariant sets of ~ with the same Conley index. We present various applications ..