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Mathematics > Dynamical Systems

arXiv:2407.08079 (math)
[Submitted on 10 Jul 2024 (v1), last revised 11 Nov 2024 (this version, v5)]

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

Authors:Wenyin Wei, Alexander Knieps, Yunfeng Liang
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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.
Comments: 9 pages, 5 figures
Subjects: Dynamical Systems (math.DS); Functional Analysis (math.FA); Optimization and Control (math.OC); Chaotic Dynamics (nlin.CD); Plasma Physics (physics.plasm-ph)
MSC classes: Primary: 34D10 Secondary: 37D10, 37N05, 37N10, 37M21, 37J40, 37J46, 39A33, 39B52, 76W05, 78M30
Cite as: arXiv:2407.08079 [math.DS]
  (or arXiv:2407.08079v5 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.2407.08079

Submission history

From: Wenyin Wei [view email]
[v1] Wed, 10 Jul 2024 22:49:31 UTC (14,178 KB)
[v2] Sat, 14 Sep 2024 11:51:48 UTC (14,178 KB)
[v3] Tue, 24 Sep 2024 15:20:03 UTC (14,178 KB)
[v4] Wed, 9 Oct 2024 21:45:57 UTC (14,234 KB)
[v5] Mon, 11 Nov 2024 18:40:28 UTC (14,326 KB)
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