2020/09/01 by Manuel Kuchelmeister, Johannes Reiff, Jörg Main +1
Computer Science · Mathematics · Physics and Astronomy · #Bifurcation #Center manifold #Classical mechanics #Hopf bifurcation #Hyperbolic set #Invariant (physics) #Invariant manifold #Manifold (fluid mechanics) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Saddle #Saddle-node bifurcation #Slow manifold #Stable manifold #physics.chem-ph #stochastic dynamics and bifurcation
paper · pdf · doi:10.1134/s1560354720050068
published as Regul. Chaotic Dyn. 29,496 (2020) · 11 pages, 6 figures
openalex publication_date 2020/09/01 · arxiv created 2020/11/08 · arxiv updated 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In chemical reactions, trajectories typically turn from reactants to products when crossing a dividing surface close to the normally hyperbolic invariant manifold (NHIM) given by the intersection of the stable and unstable manifolds of a rank-1 saddle. Trajectories started exactly on the NHIM in principle never leave this manifold when propagated forward or backward in time. This still holds for driven systems when the NHIM itself becomes time-dependent. We investigate the dynamics on the NHIM for a periodically driven model system with two degrees of freedom by numerically stabilizing the motion. Using Poincaré surfaces of section, we demonstrate the occurrence of structural changes of the dynamics, viz. , bifurcations of periodic transition state (TS) trajectories when changing the amplitude and frequency of the external driving. In particular, periodic TS trajectories with the same period as the external driving but significantly different parameters — such as mean energy — compared to the ordinary TS trajectory can be created in a saddle-node bifurcation.