On the shifts of orbits and periodic orbits under perturbation and the change of Poincar\'e map Jacobian of periodic orbits

TL;DR

This paper develops a theoretical framework using functional analysis to describe how periodic orbits and cycles shift under perturbations in dynamical systems, providing formulas for maps and flows.

Contribution

It introduces a novel approach to analyze orbit shifts using functional derivatives, enhancing understanding of system sensitivity and control.

Findings

Derived formulas for orbit shifts under perturbations for maps and flows

Provides a theoretical basis for sensitivity analysis in dynamical systems

Facilitates optimization and control of systems through orbit analysis

Abstract

Periodic orbits and cycles, respectively, play a significant role in discrete- and continuous-time dynamical systems (i.e. maps and flows). To succinctly describe their shifts when the system is applied perturbation, the notions of functional and functional derivative are borrowed from functional analysis to consider the whole system as an argument of the geometric representation of the periodic orbit or cycle. The shifts of an orbit/trajectory and periodic orbit/cycle are analyzed and concluded as formulae for maps/flows, respectively. The theory shall be beneficial for analyzing sensitivity to perturbations, and optimizing and controlling various systems.