On the shifts of stable and unstable manifolds of a hyperbolic cycle under perturbation

• DOI: 10.48550/arxiv.2407.06430
• Publication Date: 2024-01-01
• Authors: Wei, Wenyin; Hua, Jiankun; Knieps, Alexander; Liang, Yunfeng
• Keywords: Mathematics - Functional Analysis; Plasma Physics (physics.plasm-ph); FOS: Mathematics; FOS: Physical sciences; Dynamical Systems (math.DS); Mathematics - Dynamical Systems; Chaotic Dynamics (nlin.CD); Nonlinear Sciences - Chaotic Dynamics; Primary: 37D10, Secondary: 39A33, 39B52, 76W05, 78M30; Physics - Plasma Physics; Functional Analysis (math.FA)

Stable and unstable manifolds, originating from hyperbolic cycles, fundamentally characterize the behaviour of dynamical systems in chaotic regions. This letter demonstrates that their shifts under perturbation, crucial for chaos control, are computable with minimal effort using functional derivatives by considering the entire system as an argument. The shifts of homoclinic and heteroclinic orbits, as the intersections of these manifolds, are readily calculated by analyzing the movements of the intersection points.<br/>5 pages, 5 subfigures in 1 figure<br/>