Related papers: Linearization and invariant manifolds on the carry…

We present a dynamical approach to the study of unordered, attracting manifolds of retrotone maps commonly known as carrying simplices. Our approach is novel in that it uses the radial representation of unordered manifolds over the…

For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In the present paper it is shown that its convexity implies that it is a $C^1$…

We consider dissipative strongly competitive systems $\dot{x}_{i}=x_{i}f_{i}(x)$ of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface $\Sigma$, called the…

In this paper, we are concerned with studying the existence of invariant complex manifolds of two-dimensional holomorphic systems. From the geometric singular perturbation theory we know that if a slow-fast system has associated a normally…

Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold…

For a $C^1$ map $T$ from $C =[0, +\infty)^N$ to $C$ of the form $T_i(x) = x_if_i(x)$, the dynamical system $x(n) =T^n(x)$ as a population model is competitive if $\frac{\partial f_i}{\partial x_j}\leq 0$ $(i\not= j)$. A well know theorem…

Invariant manifolds are the skeleton of the chaotic dynamics in Hamiltonian systems. In Celestial Mechanics, for instance, these geometrical structures are applied to a multitude of physical and practical problems, such as to the…

We consider time-periodic competitive systems of ordinary differential equations of Kolmogorov type. However, compared with standard assumptions, we relax the regularity of the time-dependent per-capita growth rates by imposing much weaker…

This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of…

In this paper, we study the non-singular extension problem of horizontal stable fold maps. This problem asks what conditions ensure the existence of a submersion whose restriction to the boundary coincides with a given map, called a…

Multistable coupled map lattices typically support travelling fronts, separating two adjacent stable phases. We show how the existence of an invariant function describing the front profile, allows a reduction of the infinitely-dimensional…

The goal of this paper is to construct invariant dynamical objects for a (not necessarily invertible) smooth self map of a compact manifold. We prove a result that takes advantage of differences in rates of expansion in the terms of a sheaf…

There is a recent development in the carrying simplex theory for competitive maps: under some weaker conditions a map has a modified carrying simplex (one of the author's latest publications). In this paper, as one of the applications of…

We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let $f : (M,g) \to (\overline{M},\overline{g})$ be a harmonic map from a compact manifold. Instead of assuming a global nonpositivity…

The problem of linear instability of a nonlinear traveling wave in a canonical Hamiltonian system with translational symmetry subject to superharmonic perturbations is discussed. It is shown that exchange of stability occurs when energy is…

The first result in this study is a non-existence theorem for $\alpha-$harmonic mappings. Additionally, a direct connection between the $\alpha-$ harmonic and harmonic maps is made possible via conformal deformation. Second, the instability…

Fold maps are higher dimensional versions of Morse functions, which play important roles in the studies of smooth manifolds, and such general maps also have been fundamental tools in the studies of smooth manifolds by using generic maps. In…

The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its…

We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds.…