Search results for "Dynamical system"
showing 10 items of 523 documents
The case of equality in the dichotomy of Mohammadi-Oh
2017
If $n \geq 3$ and $\Gamma$ is a convex-cocompact Zariski-dense discrete subgroup of $\mathbf{SO}^o(1,n+1)$ such that $\delta_\Gamma=n-m$ where $m$ is an integer, $1 \leq m \leq n-1$, we show that for any $m$-dimensional subgroup $U$ in the horospheric group $N$, the Burger-Roblin measure associated to $\Gamma$ on the quotient of the frame bundle is $U$-recurrent.
A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature
2017
This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in \(\mathbb{R}\), \(\mathbb{C}\) and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition…
Persistence in complex systems
2022
Persistence is an important characteristic of many complex systems in nature, related to how long the system remains at a certain state before changing to a different one. The study of complex systems' persistence involves different definitions and uses different techniques, depending on whether short-term or long-term persistence is considered. In this paper we discuss the most important definitions, concepts, methods, literature and latest results on persistence in complex systems. Firstly, the most used definitions of persistence in short-term and long-term cases are presented. The most relevant methods to characterize persistence are then discussed in both cases. A complete literature r…
Two‐dimensional metric spheres from gluing hemispheres
2022
We study metric spheres (Z,dZ) obtained by gluing two hemispheres of S2 along an orientation-preserving homeomorphism g:S1→S1, where dZ is the canonical distance that is locally isometric to S2 off the seam. We show that if (Z,dZ) is quasiconformally equivalent to S2, in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves. We also show that g is bi-Lipschitz if and only if (Z,dZ) has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping h:S2→S2. Furthermore, we show that if g−1 is absolutely continuous and g admits a homeomorphic extension with exponentially integrable distortion, then (Z,dZ) …
Sobolev homeomorphic extensions onto John domains
2020
Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical Jordan-Schoenflies theorem may admit no solution - it is possible to have a boundary homeomorphism which admits a continuous $W^{1,2}$-extension but not even a homeomorphic $W^{1,1}$-extension. We prove that if the target is assumed to be a John disk, then any boundary homeomorphism from the unit circle admits a Sobolev homeomorphic extension for all exponents $p<2$. John disks, being one sided quasidisks, are of fundamental importance in Geometric Function Theory.
The Lorenz system : hidden boundary of practical stability and the Lyapunov dimension
2020
On the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global stability are estimated and the difficulties of numerically studying the birth of self-excited and hidden attractors, caused by the loss of global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov dimension along the trajectories over large time intervals is discussed. Estimating the Lyapunov dimension of attractors via the Pyragas time-delayed feedback control technique and the Leonov method is demonstrated. Taking into accoun…
System identification via optimised wavelet-based neural networks
2003
Nonlinear system identification by means of wavelet-based neural networks (WBNNs) is presented. An iterative method is proposed, based on a way of combining genetic algorithms (GAs) and least-square techniques with the aim of avoiding redundancy in the representation of the function. GAs are used for optimal selection of the structure of the WBNN and the parameters of the transfer function of its neurones. Least-square techniques are used to update the weights of the net. The basic criterion of the method is the addition of a new neurone, at a generic step, to the already constructed WBNN so that no modification to the parameters of its neurones is required. Simulation experiments and compa…
Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system
2018
In this paper, a continuous approximation to studying a class of PWC systems of fractionalorder is presented. Some known results of set-valued analysis and differential inclusions are utilized. The example of a hyperchaotic PWC system of fractional order is analyzed. It is found that without equilibria, the system has hidden attractors.
Ledrappier-Young formula and exact dimensionality of self-affine measures
2017
In this paper, we solve the long standing open problem on exact dimensionality of self-affine measures on the plane. We show that every self-affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system. In higher dimensions, under certain assumptions, we prove that self-affine and quasi self-affine measures are exact dimensional. In both cases, the measures satisfy the Ledrappier-Young formula. peerReviewed
On several notions of complexity of polynomial progressions
2021
For a polynomial progression $$(x,\; x+P_1(y),\; \ldots,\; x+P_{t}(y)),$$ we define four notions of complexity: Host-Kra complexity, Weyl complexity, true complexity and algebraic complexity. The first two describe the smallest characteristic factor of the progression, the third one refers to the smallest-degree Gowers norm controlling the progression, and the fourth one concerns algebraic relations between terms of the progressions. We conjecture that these four notions are equivalent, which would give a purely algebraic criterion for determining the smallest Host-Kra factor or the smallest Gowers norm controlling a given progression. We prove this conjecture for all progressions whose ter…