Finest Morse Decompositions for Semigroup Actions on Fiber Bundles

Let S{\mathcal{S}} be a semigroup acting on a topological space M. We study finest Morse decompositions for the action of S{\mathcal{S}} on M. This concept depends on a family of subsets of S{\mathcal{S}} . For certain semigroups and families it recovers the concept of Morse decomposition for flows and semiflows. This paper also studies the behaviour of Morse decompositions for semigroup actions on principal bundles and their associated bundles. The emphasis is put on the study of those decompositions considering their projections onto the base space and their intersections with the fibers.     

Growth Rates for Semiflows on Hausdorff Spaces

In this paper, we present a theory of vector-valued growth rates for discrete- and continuous-time semiflows on Hausdorff spaces. For a given compact flow-invariant set M and an associated growth rate, we introduce the uniform growth spectrum over M, and associated real-valued spectra via projections of the vector-valued spectrum onto one-dimensional subspaces. We show that these real-valued spectra are closed intervals if M is additionally connected. We also define the Morse spectrum associated with a growth rate by evaluating the growth rate along chains. Moreover, we relate the uniform growth spectrum to the Morse spectrum and we analyze the meaning of limit sets for the long-time behavior of growth rates.     

Bifurcations and Attractor-Repeller Splittings of Non-Saddle Sets

This paper is devoted to the study of some aspects of the stability theory of flows. In particular, we study Morse decompositions induced by non-saddle sets, including their corresponding Morse equations, attractor-repeller splittings of non-saddle sets and bifurcations originated by implosions of the basin of attraction of asymptotically stable fixed points. We also characterize the non-saddle sets of the plane in terms of the Euler characteristic of their region of influence.     

A Uniform Exponential Spectrum for Linear Flows on Vector Bundles

For linear flows on vector bundles we define a uniform exponential spectrum. For a compact invariant set for the projected flow we obtain this spectrum by taking all accumulation points for the time tending to infinity of the union over the finite time exponential growth rates for all initial values in this set. Using direct arguments we show that for a connected compact invariant set this spectrum is a closed interval whose boundary points are Lyapunov exponents. For a compact invariant set on which the flow is chain transitive we show that this spectrum coincides with the Morse spectrum. In particular, this approach admits a straightforward analytic proof for the regularity and continuity properties of the Morse spectrum without using cohomology or ergodicity results.     

Semiflows on Topological Spaces: Chain Transitivity and Semigroups

This paper studies semiflows on topological spaces. A concept of chain recurrence, based on families of coverings, is introduced and related to Morse decomposition. The chain transitive components are studied via semigroup theory by the introduction of the shadowing semigroups associated to a semiflow.     

The attractor of the scalar reaction diffusion equation is a smooth graph

For the scalar reaction diffusion equation with Dirichlet boundary conditions, it is proved that its maximal compact attractor is the graph of a C¹ function from a subset with nonempty interior of a subspace of the state space the dimension of which is equal to the maximal Morse index of the equilibria of the equation.     

Ergodic theorem for infinite iterated function systems

1IntroductionandProblem IteratedFunctionSystems(IFS)theorycanbesaidtobethecontinuationanddevelopment ofdynamicalsystemtheory.DynamicalsystemstheorydealswithiterationofonemapbutIFS theorydealswithiterationofmanymaps. IFStheory’srootwasveryearlybutthebeginningofactivedevelopmentwasHutchinson’s paper(1981).Heresearchedselfsimilarityoffractalsetsusingsystemoffinitenumberofsimilar contractionmapsofRn.Barnsleycalledafinitesetofcontractionmappsasaniteratedfunction systemsandsystemizedIFStheo…     

On the embedding and compact properties of finite element spaces

In this paper, the generalized Sobolev embedding theorem and the generalized Rellich-Kondrachov compact theorem for finite element spaces with multiple sets of functions are established. Specially, they are true for nonconforming, hybrid and quasi-conforming element spaces with certain conditions.     

Boiling heat transfer enhancement of water on tubes in compact in-line bundles

In desalinization devices and some heat exchangers making use of low-quality heat energy, both wall temperatures and wall heat fluxes of the heated tubes are generally quite low; hence they cannot cause boiling in flooded tube-bundle evaporators with common large tube spacing. However, when the tube spacing is very small, the incipient boiling in restricted spaces can generate and results in higher heat transfer than that of pool boiling at the same heat flux. This study investigated experimentally the effects of tube spacing, positions of tubes and test pressures on the boiling heat transfer of water in restricted spaces of the compact in-line bundles consisting of smooth horizontal tubes. The experimental results show that tube spacing and tube position have significant effects on the boiling heat transfer in a compact tube bundle. There is an optimum tube spacing that provides the largest heat transfer coefficient at the same heat flux.     

FIXED POINT THEOREMS FOR MAPPINGS SATISFYING IMPLICIT RELATION ON TWO COMPLETE AND COMPACT METRIC SPACES

First, the implicit relations were given. A common fixed point theorem was proved for two mappings satisfying implicit relation functions. A further fixed point theorem was proved for mappings satisfying implicit relation functions on two compact metric spaces.     

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier- Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.     

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.     

Binary Decompositions for Planar N-Body Problems and Symmetric Periodic Solutions

A binary decomposition for a system of N masses is a way of treating the system as binaries with the total action exactly the same as that of the original system. By considering binary decompositions, we are able to provide effective lower-bound estimates for the action of collision paths in several spaces of symmetric loops. As applications, we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the N-body problem.     

Central configurations of the N-body problem via equivariant Morse theory

In this paper we use the equivariant Morse theory to give an estimate of the minimal number of central configurations in the N-body problem in ³. In the case of equal masses we prove that the planar central configurations are saddle points for the potential energy. From this we deduce the presence of non-planar central configurations, for every N 4.The principal difficulty in applying Morse theory is that the potential function is defined on a manifold on which the group O(3) does not act freely. To avoid this problem the equivariant cohomology functor is applied in order to obtain the Morse inequalities.     

Morse Decompositions and Spectra on Flag Bundles

For linear flows on vector bundles, the chain recurrent components of the induced flows on flag bundles are described and a corresponding Morse spectrum is constructed.     

An Estimate for the Morse Index of a Stokes Wave

Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two-dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler–Lagrange equation of a certain functional; this allows one to define the Morse index of a Stokes wave. It is well known that if the Morse indices of the elements of a set of non-singular Stokes waves are bounded, then none of them is close to a singular one. The paper presents a quantitative variant of this result.     

Nontrivial Solutions for a Semilinear Dirichlet Problem with Nonlinearity Crossing Multiple Eigenvalues

In this paper we prove that a semilinear elliptic boundary value problem has at least three nontrivial solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues of the Laplacian and all solutions are nondegenerate. A pair are of one sign (positive and negative, respectively). The one sign solutions are of Morse index less than or equal to 1 and the third solution has Morse index greater than or equal to 2. Extensive use is made of the mountain pass theorem, and Morse index arguments of the type Lazer–Solimini (see Lazer and Solimini, Nonlinear Anal. 12(8), 761–775, 1988). Our result extends and complements a theorem of Cossio and Veléz, Rev. Colombiana Mat. 37(1), 25–36, 2003.AMS Subject classification: 35J20; 35J25; 35J60.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Symbolic Image, Hyperbolicity, and Structural Stability

The aim of the paper is substantiation of a constructive method for verification of hyperbolicity and structural stability of discrete dynamical systems. The main tool here is a symbolic image which is a directed graph constructed by a finite covering of the projective bundle. Hyperbolicity is tested by calculation of the Morse spectrum (the limit set of Lyapunov exponents of pseudo trajectories) which can be found for a given accuracy by the symbolic image [24]. If the Morse spectrum does not contain 0, then the chain recurrent set is hyperbolic and the system is Ω-stable. Thus, the symbolic image gives an opportunity to verify these properties. A diffeomorphism f is shown to be structurally stable if and only if the Morse spectrum does not contain 0 and for the complementary differential there is no connection CR ⁺→CR ^? on the protective bundle. These conditions are verified by an algorithm based on the symbolic image of the complementary differential.     

Minimal Spaces with Cyclic Group of Homeomorphisms

There are two main subjects in this paper. (1) For a topological dynamical system \((X,T)\) we study the topological entropy of its “functional envelopes” (the action of \(T\) by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of \(X\)). In particular we prove that for zero-dimensional spaces \(X\) both entropies are infinite except when \(T\) is equicontinuous (then both equal zero). (2) We call Slovak space any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems \((X,T)\) with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.