Embedding Information onto a Dynamical System

TL;DR

This paper explores how sequences can be embedded into dynamical systems as attractive solutions, providing insights into state space models and the effects of noise perturbations on attractors.

Contribution

It introduces a method to embed arbitrary sequences into nonautonomous dynamical systems with topological conjugacy, extending understanding of state space models and noise effects.

Findings

Sequences can be embedded as attractive solutions in nonautonomous systems.

The embedding preserves topological conjugacy between sequences and solutions.

Perturbations by noise affect the dynamics of attractors in specific, describable ways.

Abstract

The celebrated Takens' embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In this paper, we show how an arbitrary sequence can be mapped into another space as an attractive solution of a nonautonomous dynamical system. Such mapping also entails a topological conjugacy and an embedding between the sequence and the attractive solution spaces. This result is not a generalization of Takens embedding theorem but helps us understand what exactly is required by discrete-time state space models widely used in applications to embed an external stimulus onto its solution space. Our results settle another basic problem concerning the perturbation of an autonomous dynamical system. We describe what exactly happens to the dynamics when exogenous noise perturbs continuously a local irreducible attracting set (such as a stable fixed point) of a discrete-time autonomous dynamical system.