We use homotopy index and monotonicity techniques to study the connecting orbits of systems of two competing species equations with diffusion and large interaction. We also use earlier work of Zhitao Zhang and the author on the dynamics of this system.

In his seminal paper of 1967 on disjointness in topological dynamics and ergodic theory H. Furstenberg started a systematic study of transitive dynamical systems. In recent years this work served as a basis for a broad classification of dynamical systems by their recurrence properties. In this paper I describe some aspects of this new theory and its connections with combinatorics, harmonic analysis and the theory of topological groups.

In this paper, we study a class of inhomogeneous Neumann boundary value problems on a compact Riemannian manifold with boundary where indefinite and critical nonlinearities are included. Applying the fibering approach we introduce a new and, in some sense, more general variational approach to these problems. Using this idea we prove new results on the existence and multiplicity of positive solutions.

Global existence of regular solutions to the Navier-Stokes equations in a bounded cylindrical domain without the axis of symmetry and with boundary slip conditions is proved. We showed the existence of solutions without restrictions on the magnitude of the initial velocity assuming only that the $L_2$-norms of the angular derivative of the cylindrical components of the initial velocity and the external force are sufficiently small. To prove global existence some decay estimates on the external force are imposed.

The method of invariant manifolds, now called the Hadamard-Perron Theorem, was originally developed by Lyapunov, Hadamard and Perron for time-independent maps and differential equations at a hyperbolic fixed point. It was then extended from hyperbolic to non-hyperbolic systems, from time-independent and finite-dimensional to time-dependent and infinite-dimensional equations. The generalization of an invariant manifold for a discrete dynamical system (mapping) to a time-variant difference equation is called an invariant fiber bundle. While in the hyperbolic case the smoothness of the invariant fiber bundles is easily obtained with the contraction principle, in the non-hyperbolic situation the smoothness depends on a spectral gap condition, is subtle to prove and proofs were given under various assumptions by basically three different approaches, so far: (1) A lemma of Henry, (2) the fiber-contraction theorem, or (3) fixed point theorems for scales of embedded Banach spaces.

In this paper we present a new self-contained and basic proof of the smoothness of invariant fiber bundles which relies only on Banach's fixed point theorem. Our result extends previous versions of the Hadamard-Perron Theorem and generalizes it to the time-dependent, not necessarily hyperbolic, infinite-dimensional, non-invertible and parameter-dependent case. Moreover, we show by an example that our gap-condition is sharp.

We extend to normalized systems several properties of commuting systems proved in [M. Sabatini, Dynamics of commuting systems on two-dimensional manifolds, Ann. Mat. Pura Appl. CLXXIII (IV) (1997), 213–232]. A rough classification of the dynamics induced by normalized vector fields on two-dimensional compact connected oriented manifolds is given.

The paper is devoted to infinite dimensional discrete operators that can be considered as a difference analog of differential equations on the whole axis. We obtain a necessary and sufficient condition in order for the linear operator to be normally solvable. Topological degree for nonlinear operators is constructed.

We prove some fixed point theorems for the Hausdorff continuous multivalued mappings with equilocally connected values in dimension $n - 1$ or $n - 2$ on $n$-dimensional discs and closed manifolds.