arXiv research

The unknotting number of a positive braid with n strands and k intersections is known to be equal to (k-n+1)/2. We consider Lorenz knots (which are positive braids) and, using a different method, find their unknotting numbers in terms of their positions on the Lorenz attractor.

In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We also analyze some natural Morse decompositions of the global attractor of the syst…

New risk models use chaotic attractors to predict extreme events.

Low-connectivity reservoirs outperform standard designs in chaotic system forecasting.

We analyze the time series of four major cryptocurrencies (Bitcoin, Ethereum, Litecoin, and Ripple) before the digital market crash at the end of 2017 - beginning 2018. We introduce a methodology that combines topological data analysis with a machine learning technique -- $k$-means clustering -- in order to automatical…

Enhances forecasting of complex systems using FKMD.

New satellite knots counter a conjecture about Lorenz knots.

Paper analyzes coexisting hidden and self-excited attractors in an economic system.

Characterizes knotted toroidal sets as attractors in 3D.

We describe the Williams zeta functions and the twist zeta functions of sub-Lorenz templates generated by renormalizable Lorenz maps, in terms of the corresponding zeta-functions of the sub-Lorenz templates generated by the renormalized map and by the map that determines the renormalization type.

We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots code…

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. T…

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…

This work uses a scalable approach to identify partially observed nonlinear systems.

This is a review article on Lorenz knots.

This paper classifies expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.

Proposes a new method combining Reservoir Computing and Normalizing Flow for predicting stochastic dynamical systems.

This article is a survey on Lorenz knots. We describe the original construction, prove several classical properties, in particular the fact that the closure of a positive braid is a fibered knot, and describe Ghys'correspondance between modular knots and Lorenz knots. We also prove two new properties, namely that follo…

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit form…

Study of Lorenz links and T-links, showing equivalence and unique presentations.

We exhibit low-dilatation families of surface homeomorphisms among monodromies of Lorenz knots.

A graphical model is a structured representation of the data generating process. The traditional method to reason over random variables is to perform inference in this graphical model. However, in many cases the generating process is only a poor approximation of the much more complex true data generating process, leadi…

We introduce a mathematical model on the dynamics of demand and supply incorporating collectability and saturation factors. Our analysis shows that when the fluctuation of the determinants of demand and supply is strong enough, there is chaos in the demand-supply dynamics. Our numerical simulation shows that such a cha…

The paper models star dynamics using Ricci flow and Perelman entropy, revealing chaotic behavior.

The paper studies dimensions of attractors for modified Leray-alpha equation on various surfaces.

Study bounds topological entropy of toroidal attractors.

In paper "A new twist on Lorenz links" (Journal of Topology 2(2009), 227-248) Joan Birman and Ilya Kofman prove the coincidence of the class of Lorenz links and the class of twisted links. The proof in that work is algebraic. We will identify this class in terms of grid diagrams and provide a transparent geometric argu…

We prove a theorem on structural stability of smooth attractor-repellor endomorphisms of compact manifolds, with singularities. By attractor-repellor, we mean that the non-wandering set of the dynamics $f$ is the disjoint union of a repulsive compact subset with a hyperbolic attractor on which $f$ acts bijectively. The…

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $Ω(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expandi…

Computed linking number of modular knots and Lorenz links.

In this article the Lorenz dynamical system is revived and revisited and the current state of the art results for one step ahead forecasting for the Lorenz trajectories are published. Multitask learning is shown to help learning the hard to learn z trajectory. The article is a reflection upon the evolution of neural ne…

A central challenge faced by memory systems is the robust retrieval of a stored pattern in the presence of interference due to other stored patterns and noise. A theoretically well-founded solution to robust retrieval is given by attractor dynamics, which iteratively clean up patterns during recall. However, incorporat…

New findings on T-links derived from torus links.

The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.

New satellite knots found that can't be represented by positive braids with full twists.

A new method for analyzing high-dimensional time-series data using deep neural networks.

During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic p…

We show that the zeroes of the Alexander polynomial of a Lorenz knot all lie in some annulus whose width depends explicitly on the genus and the braid index of the considered knot.

Improved algorithm for modular links provides upper volume bounds.

In human perception and cognition, a fundamental operation that brains perform is interpretation: constructing coherent neural states from noisy, incomplete, and intrinsically ambiguous evidence. The problem of interpretation is well matched to an early and often overlooked architecture, the attractor network---a recur…

In this paper we focus on compacta $K \subseteq \mathbb{R}^3$ which possess a neighbourhood basis that consists of nested solid tori $T_i$. We call these sets toroidal. In \cite{hecyo1} we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory. Here we introduce the se…

New method reconstructs hidden dynamics from low-dimensional time series.

Study analyzes Echo State Network parameters for Rossler attractor dynamics.

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile $\mathbb{R}^3$. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to hig…

In this paper we consider the realization of DE attractors by self-diffeomorphisms of manifolds. For any expanding self-map $φ:M\to M$ of a connected, closed $p$-dimensional manifold $M$, one can always realize a $(p,q)$-type attractor derived from $φ$ by a compactly-supported self-diffeomorphsm of $\RR^{p+q}$, as long…

Introduction: Machine learning provides fundamental tools both for scientific research and for the development of technologies with significant impact on society. It provides methods that facilitate the discovery of regularities in data and that give predictions without explicit knowledge of the rules governing a syste…

As a first step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as th…

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? T…