arXiv research

Convolutional attractor nets improve image completion and super-resolution.

Study analyzes Echo State Network parameters for Rossler attractor dynamics.

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…

Learning three data points can generate all types of periodic orbits in a neural network.

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…

Paper refines RNN training by analyzing smoothness and attractors.

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

Deep neural networks can store and recall data efficiently.

Reverse engineered RNNs reveal line attractor dynamics for sentiment classification.

Characterizes knotted toroidal sets as attractors in 3D.

GRUs exhibit diverse dynamical behaviors but cannot mimic continuous attractors.

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…

Machine learning classifiers are often trained to recognize a set of pre-defined classes. However, in many applications, it is often desirable to have the flexibility of learning additional concepts, with limited data and without re-training on the full training set. This paper addresses this problem, incremental few-s…

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

Neural networks can model chaos efficiently by becoming geometrically chaotic.

Deep learning methods improve overlapping speaker separation across languages and noise.

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 characterizes toroidal sets as attractors for flows and homeomorphisms of R^3.

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

Study bounds topological entropy of toroidal attractors.

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…

It has been argued in the past that high-dimensional neural networks do not exhibit local minima capable of trapping an optimisation algorithm. However, the relationship between loss surface modality and the neural architecture parameters, such as the number of hidden neurons per layer and the number of hidden layers, …

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

Meta-learning approach for fast and compressive energy-based memory models.

ERDMD discovers sparse, nonuniformly timed DMD models from chaotic attractors.

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

ESNs with transfer learning predict long-term chaotic patterns in spatiotemporal dynamical systems.

New risk models use chaotic attractors to predict extreme events.

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…

New theory explains how chaotic training improves neural network generalization.

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…

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…

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…

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

Unified Bayesian framework predicts cryptocurrency market dynamics and volatility.

In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to \mathbb{R}^d$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index…

New study shows min-max algorithms can converge to non-stationary points.

We analyze the dynamical properties of a tetrahedron transformation on the space of non-degenerate tetrahedra which can be identified with the non-compact globally symmetric $8$-dimensional space $\mbox{Sl}(3,\mathbb{R}) / \mbox{So}(3,\mathbb{R})$. We establish the existence of a local attractor which coincides with th…

An iterated function system $Φ$ consisting of contractive similarity mappings has a unique attractor $F \subseteq \mathbb{R}^d$ which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling $\mathcal{T}$ of the con…

The paper studies bifurcations in discrete dynamical systems on manifolds.

Researchers analyze how RNNs solve intent detection tasks using dynamical systems theory.

Let $M$ be a manifold or (more generally) a locally compact, metrizable ANR. If $K$ is an attractor for a flow in $M$, with basin of attraction $\mathcal{A}(K)$, it is well known that the inclusion $i : K \subseteq \mathcal{A}(K)$ is always a shape equivalence. In this paper we investigate to what extent this generaliz…

This paper will be splited into two papers and submited later.

Static spacetimes are stable attractors in a flow equation.

We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. The…

RNNs classify text by accumulating evidence on a low-dimensional manifold.