On the shifts of orbits and periodic orbits under perturbation and the change of Poincaré map Jacobian of periodic orbits

2024/07/10 by Wei, Wenyin, Knieps, Alexander, Liang, Yunfeng
#37J40 #37J46 #37M21 #37N05 #37N10 #39A33 #39B52 #76W05 #78M30 #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Optimization and Control (math.OC) #Plasma Physics (physics.plasm-ph) #Primary: 34D10 Secondary: 37D10

paper · doi:10.48550/arxiv.2407.08079

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.

Related

• On the shifts of stable and unstable manifolds of a hyperbolic cycle under perturbation
• The Period adding and incrementing bifurcations: from rotation theory to applications
• A proof of the C^r closing lemma and stability conjecture
• On the bifurcation of periodic orbits
• Perturbation theory of dynamical systems