Embedding Information onto a Dynamical System
This work provides theoretical insights for discrete-time state space models in applications, addressing foundational issues in dynamical systems theory, though it is incremental relative to Takens' theorem.
The paper tackles the problem of embedding arbitrary sequences into the solution space of a nonautonomous dynamical system, establishing topological conjugacy and embedding properties, and it resolves a basic problem about how exogenous noise perturbs local attracting sets in discrete-time autonomous systems.
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.