Chaos on compact manifolds: Differentiable synchronizations beyond the Takens theorem

TL;DR

This paper demonstrates that a broad class of fading memory state-space systems on compact manifolds can achieve continuously differentiable synchronizations, enhancing chaotic attractor analysis and surpassing previous limitations.

Contribution

It extends the theory of differentiable generalized synchronizations to a wider class of systems on compact manifolds, providing new tools for chaos analysis.

Findings

Large class of systems yield differentiable synchronizations

Improves understanding of chaotic attractor reconstruction

Enhances methods for forecasting chaotic systems

Abstract

This paper shows that a large class of fading memory state-space systems driven by discrete-time observations of dynamical systems defined on compact manifolds always yields continuously differentiable synchronizations. This general result provides a powerful tool for the representation, reconstruction, and forecasting of chaotic attractors. It also improves previous statements in the literature for differentiable generalized synchronizations, whose existence was so far guaranteed for a restricted family of systems and was detected using H\"older exponent-based criteria.