Extension of Chronological Calculus for Dynamical Systems on Manifolds

TL;DR

This paper extends the Chronological Calculus to handle less smooth and infinite-dimensional dynamical systems, enabling new computational methods and proving a variant of the Chow-Rashevskii theorem.

Contribution

It introduces a generalized Chronological Calculus applicable to $C^m$-smooth and infinite-dimensional manifolds, simplifying computations without Fréchet calculus.

Findings

Extended calculus handles $C^m$-smooth systems and infinite-dimensional manifolds.

Provides a computational tool avoiding Fréchet space calculus.

Proves a variant of the Chow-Rashevskii theorem for infinite-dimensional manifolds.

Abstract

We propose an extension of the Chronological Calculus, developed by Agrachev and Gamkrelidze for the case of $C^\infty$-smooth dynamical systems on finite-dimensional $C^\infty$-smooth manifolds, to the case of $C^m$-smooth dynamical systems and infinite-dimensional $C^m$-manifolds. Due to a relaxation in the underlying structure of the calculus, this extension provides a powerful computational tool without recourse to the theory of calculus in Fr\'echet spaces required by the classical Chronological Calculus. In addition, this extension accounts for flows of vector fields which are merely measurable in time. To demonstrate the utility of this extension, we prove a variant of Chow-Rashevskii theorem for infinite-dimensional manifolds.