Dynamics and bifurcations on the normally hyperbolic invariant manifold of a periodically driven system with rank-1 saddle
arXiv:2011.04042 · doi:10.1134/S1560354720050068
Abstract
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, \emph{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.
arXiv comment11 pages, 6 figures
References in corpus (23)
- SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
- Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
- Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum
- The hidden geometry of ocean flows
- The Transition State in a Noisy Environment
- Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry
- Stochastic Transition States: Reaction Geometry amidst Noise
- Homoclinic chaos in the dynamics of a general Bianchi IX model
- Identifying reactive trajectories using a moving transition state
- Stability and Decay Rates of Non-Isotropic Attractive Bose-Einstein Condensates
- Obtaining time-dependent multi-dimensional dividing surfaces using Lagrangian descriptors
- Chemical reactions induced by oscillating external fields in weak thermal environments
- Transition state trajectory stability determines barrier crossing rates in chemical reactions induced by time-dependent oscillating fields
- Invariant Manifolds and Rate Constants in Driven Chemical Reactions
- Persistence of transition state structure in chemical reactions driven by fields oscillating in time
- Binary contraction method for the construction of time-dependent dividing surfaces in driven chemical reactions
- Influence of external driving on decays in the geometry of the LiCN isomerization
- Phase-space resolved rates in driven multidimensional chemical reactions
- Transition state theory for wave packet dynamics. II. Thermal decay of Bose-Einstein condensates with long-range interaction
- Transition states and thermal collapse of dipolar Bose-Einstein condensates
- Neural network approach for the dynamics on the normally hyperbolic invariant manifold of periodically driven systems
- Transition state theory for wave packet dynamics. I. Thermal decay in metastable Schrödinger systems
- Dynamics and decay rates of a time-dependent two-saddle system