Related papers: Attractiveness of Invariant Manifolds
We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The problem of reduced description is studied as a problem of constructing the slow…
The beacon model is a recent paradigm for guiding the trajectory of messages or small robotic agents in complex environments. A beacon is a fixed point with an attraction pull that can move points within a given polygon. Points move…
While stability analysis is a mainstay for control science, especially computing regions of attraction of equilibrium points, until recently most stability analysis tools always required explicit knowledge of the model or a high-fidelity…
In this paper, several results concerning attraction and asymptotic stability in the large of nonlinear ordinary differential equations are presented. The main result is very simple to apply yielding a sufficient condition under which the…
We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce…
We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $\phi$. For the normal case, we prove that a $\phi$-invariant submanifold tangent to a Reeb vector field…
Consider a dynamical system $T:\mathbb{T}\times \mathbb{R}^{d} \rightarrow \mathbb{T}\times \mathbb{R}^{d} $ given by $ T(x,y) = (E(x), C(y) + f(x))$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of…
A classical theorem in conformal geometry states that on a manifold with non-positive Yamabe invariant, a smooth metric achieving the invariant must be Einstein. In this work, we extend it to the singular case and show that in all…
Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov-Schmidt reduction for…
One of basic difficulties of machine learning is handling unknown rotations of objects, for example in image recognition. A related problem is evaluation of similarity of shapes, for example of two chemical molecules, for which direct…
We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally…
In Hamiltonian systems subjected to periodic perturbations the stable and unstable manifolds of the unstable periodic orbits provide the dynamical "skeleton" that drives the mixing process and bounds the chaotic regions of the phase space.…
Recordings of increasingly large neural populations have revealed that the firing of individual neurons is highly coordinated. When viewed in the space of all possible patterns, the collective activity forms non-linear structures called…
A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with…
In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the…
Invariant manifolds of unstable periodic orbits organize the dynamics of chaotic orbits in phase space. They provide insight into the mechanisms of transport and chaotic advection and have important applications in physical situations…
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 construct an open set of endomorphisms of an arbitrary two-dimensional manifold which have attractors and non-wandering sets with non-invariant interior. This is a notable contrast to the properties of diffeomorphisms, where the interior…
In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an…
We show that the Makar-Limanov invariant of the cylinder over the Koras-Russell cubic affine threefold is trivial. This means that regular functions which are invariant under all algebraic actions of the additive group on this variety are…