arXiv research

New risk models use chaotic attractors to predict extreme events.

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

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

The use of artificial neural networks as models of chaotic dynamics has been rapidly expanding. Still, a theoretical understanding of how neural networks learn chaos is lacking. Here, we employ a geometric perspective to show that neural networks can efficiently model chaotic dynamics by becoming structurally chaotic t…

New theory explains how chaotic training improves neural network generalization.

Investigates chaotic financial time series with monthly contributions and devaluation.

Bayesian framework detects symmetries in chaotic dynamical systems.

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

We have carried out simulations of a financial model of the firm to analyse the validity of the concept of Trade on Equity in dynamics. The results exhibit the ability of the borrowing policy connected to a cautious dividend distribution to inject chaos into the profit motion. The 3D system built with the van der Pol's…

New method learns chaotic dynamics from single noisy trajectory.

We explore the hyperparameter space of reservoir computers used for forecasting of the chaotic Lorenz '63 attractor with Bayesian optimization. We use a new measure of reservoir performance, designed to emphasize learning the global climate of the forecasted system rather than short-term prediction. We find that optimi…

New insights into how large learning rates affect transformer training dynamics.

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

The OGY method is one of control methods for a chaotic system. In the method, we have to calculate a stabilizing periodic orbit embedded in its chaotic attractor. Thus, we cannot use this method in the case where a precise mathematical model of the chaotic system cannot be identified. In this case, the delayed feedback…

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…

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

In Part II of this paper, we concentrate our analysis on the price dynamical model with the moving average rules developed in Part I of this paper. By decomposing the excessive demand function, we reveal that it is the interplay between trend-following and contrarian actions that generates the price chaos, and give par…

Characterizes knotted toroidal sets as attractors in 3D.

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…

A method models nonlinear dynamics from data using barycentric coordinates and memory.

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

The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.

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 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…

There is by now a large consensus in modern monetary policy. This consensus has been built upon a dynamic general equilibrium model of optimal monetary policy as developed by, e.g., Goodfriend and King (1997), Clarida et al. (1999), Svensson (1999) and Woodford (2003). In this paper we extend the standard optimal monet…

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…

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…

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

Reservoir Computing enhances climate predictability studies.

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…

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…

Study chaotic behavior in homeomorphism groups of countable products of spaces.

Study analyzes Echo State Network parameters for Rossler attractor dynamics.

TSSC images enhance chaotic signal classification using ConvNets.

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…

Reservoir computing predicts chaotic systems for long horizons with sparse updates.

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…

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

Bayesian ANN method predicts chaotic systems with uncertainty.

Unified Bayesian framework predicts cryptocurrency market dynamics and volatility.

Panda predicts chaotic systems without retraining, showing emergent properties.