Related papers: On Conley's Fundamental Theorem of Dynamical Syste…

We investigate the (linearized) Morse index of solutions to Hamiltonan systems, with a focus on convex Hamiltonians functions and sign-changing radial solutions. For strongly coupled systems, we describe the profile of the radial solutions…

We study the problem of detecting the structure of a complex dynamical system described by a set of deterministic differential equation that contains a Hamiltonian subsystem, without any information on the explicit form of evolution laws.…

We offer streamlined proofs of fundamental theorems regarding the index theory for partial self-maps of an infinite set that are bijective between cofinite subsets.

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…

We establish a simple generalization for the famous theorem of Morley about trisectors in triangle with a purely synthetic proof using only angle chasing and similar triangles. Furthermore, based on the converse construction, another simple…

We extend the theory of transience to general dynamical systems with no Markov structure assumed. This is linked to the theory of phase transitions. We also provide examples of new kinds of transient behaviour.

In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane.

For autonomous Tonelli systems on $\R^n$, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new…

We give a necessary and sufficient condition for strong stability of low dimensional Hamiltonian systems, in terms of the iterates of a closed orbit and the Conley-Zehnder index. Applications to Mathieu equation and stable harmonic…

The aim of this paper is to explore the possibilities of Conley index techniques in the study of heteroclinic connections between finite and infinite invariant sets. For this, we remind the reader of the Poincar\'e compactification: this…

The Szymczak functor is a tool used to construct the Conley index for dynamical systems with discrete time. We present an algorithmizable classification of isomorphism classes in the Szymczak category over the category of finite sets with…

A dynamic iteration scheme for linear infinite-dimensional port-Hamiltonian systems is proposed. The dynamic iteration is monotone in the sense that the error is decreasing, it does not require any stability condition and is in particular…

This paper gives the definition of the Conley index for a piecewise continuous map, which is only well defined on compatible isolating neighborhoods with Wazewski property slightly weaker than continuous situation.

This is a note on some results of the central limit theorem for deterministic dynamical systems. First, we give the central limit theorem for martingales, which is a main tool. Then we give the main results on the central limit theorem in…

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…

The purpose of this work is to develop a version of Forman's discrete Morse theory for simplicial complexes, based on internal strong collapses. Classical discrete Morse theory can be viewed as a generalization of Whitehead's collapses,…

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between…

Consider the infinite dimensional hyperbolic dynamical system provided by the (forward) heat semi-flow on the loop space of a closed Riemannian manifold M. We use the recently discovered backward {\lambda}-Lemma and elements of Conley…

A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special…

We study the theoretical properties of the generalized dynamic principal components introduced in Pe\~na and Yohai (2016). In particular, we prove that when the data follows a dynamic factor model, the reconstruction provided by the…