Dynamics and Bifurcations on the Normally Hyperbolic Invariant Manifold of a Periodically Driven System with Rank-1 Sadd
- PDF / 986,502 Bytes
- 12 Pages / 612 x 792 pts (letter) Page_size
- 41 Downloads / 222 Views
Dynamics and Bifurcations on the Normally Hyperbolic Invariant Manifold of a Periodically Driven System with Rank-1 Saddle Manuel Kuchelmeister1* , Johannes Reiff1** , J¨ org Main1*** , and Rigoberto Hernandez2, 3**** 1
Institut f¨ ur Theoretische Physik I, Universit¨ at Stuttgart, 70550 Stuttgart, Germany 2 Department of Chemistry, Johns Hopkins University, Baltimore, 21218 Maryland, United States 3 Departments of Chemical & Biomolecular Engineering, and Materials Science and Engineering, Johns Hopkins University, Baltimore, 21218 Maryland, United States Received July 13, 2020; revised August 29, 2020; accepted September 09, 2020
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 timedependent. We investigate the dynamics on the NHIM for a periodically driven model system with two degrees of freedom by numerically stabilizing the motion. Using Poincar´e 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. MSC2010 numbers: 37D05, 37G15, 37J20, 37M05, 65P30 DOI: 10.1134/S1560354720050068 Keywords: transition state theory, rank-1 saddle, normally hyperbolic invariant manifold, stroboscopic map, bifurcation
1. INTRODUCTION Transition state theory (TST) [3, 6, 14, 16, 17, 20, 31, 34, 43, 44, 46, 48, 51, 56, 64– 66, 71] is well established for the computation of rates in systems with a rank-1 saddle. In these systems, two different states — viz., reactants and products — are classified and separated by an appropriately chosen dividing surface (DS). TST uses the flux through the DS to determine the rate of a chemical reaction or a similar process. It has been applied in a broad range of problems in a broad range of fields including atomic physics [22], solid state physics [21], cluster formation [33, 35], diffusion dynamics [63, 69], cosmology [45], celestial mechanics [23, 70], and Bose – Einstein condensates [18, 19, 25, 28, 29]. For systems which are time-dependently driven, e.g., by an oscillating external field, the situation becomes more challenging because the DS itself becomes time-dependent and depends nontrivially on the moving saddle of the *
E-mail: E-mail: *** E-mail: **** E-mail: **
[email protected] [email protected] [email protected] [email protected]
496
DYNAMICS AN
Data Loading...
