The Role of Time-Dependent Phase Space Structures in Reaction Dynamics and the No-Recrossing Property of Dividing Surfaces

2021 ◽  
Vol 31 (04) ◽  
pp. 2150064
Author(s):  
Cate Mandell ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Time Dependence ◽  
Reaction Dynamics ◽  
Time Dependent ◽  
Space Structures ◽  
Stable And Unstable Manifolds ◽  
Normally Hyperbolic Invariant Manifold ◽  
Unstable Manifolds ◽  
Dividing Surface

We analyze benchmark models for reaction dynamics associated with a time-dependent saddle point. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold; its time-dependent stable and unstable manifolds; and a time-dependent dividing surface that has the no-recrossing property. Consideration of the time dependence of these phase space structures is necessary in order to precisely capture reacting and nonreacting trajectories. Moreover, we show that a time-dependent dividing surface is necessary in order to eliminate recrossing in the time-dependent setting. In other words, if the dividing surface is not time-dependent, recrossing may occur.

scholarly journals Tilting and Squeezing: Phase Space Geometry of Hamiltonian Saddle-Node Bifurcation and its Influence on Chemical Reaction Dynamics

2020 ◽  
Vol 30 (04) ◽  
pp. 2030008 ◽  
Author(s):  
Víctor J. García-Garrido ◽  
Shibabrat Naik ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Potential Energy ◽  
Invariant Manifolds ◽  
Reaction Dynamics ◽  
Potential Energy Function ◽  
Space Structures ◽  
Stable And Unstable Manifolds ◽  
Saddle Node Bifurcation ◽  
Saddle Node

In this article, we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete “phase space tomography” by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.

An algorithm for computing phase space structures in chemical reaction dynamics using Voronoi tessellation

2021 ◽  
pp. 133047
Author(s):  
Yuta Mizuno ◽  
Mikoto Takigawa ◽  
Saki Miyashita ◽  
Yutaka Nagahata ◽  
Hiroshi Teramoto ◽  
...  
Keyword(s):  
Phase Space ◽  
Chemical Reaction ◽  
Voronoi Tessellation ◽  
Reaction Dynamics ◽  
Space Structures ◽  
Chemical Reaction Dynamics

scholarly journals Role of phase space structures in collisionless drift wave turbulence and impact on transport modeling

2017 ◽  
Vol 57 (7) ◽  
pp. 072006 ◽  
Author(s):  
Y. Kosuga ◽  
S.-I. Itoh ◽  
P.H. Diamond ◽  
K. Itoh ◽  
M. Lesur
Keyword(s):  
Phase Space ◽  
Transport Modeling ◽  
Space Structures ◽  
Wave Turbulence ◽  
Drift Wave Turbulence ◽  
Drift Wave

scholarly journals Phase Space Analysis of Relaxation Dynamics of Isoscalar Giant Monopole Resonances

2019 ◽  
Vol 14 ◽  
pp. 131
Author(s):  
P. K. Papachristou ◽  
E. Mavrommatis ◽  
V. Constantoudis ◽  
F. Κ. Diakonos ◽  
J. Wambach
Keyword(s):  
Phase Space ◽  
Collective Excitation ◽  
Classical Model ◽  
Space Structures ◽  
Relaxation Dynamics ◽  
Nuclear Dynamics ◽  
Independent Particle ◽  
The Common ◽  
Broad Region

A classical model based on the independent particle approach to the nuclear dynamics is used to study the influence of the phase space structures on the onebody dissipation of isoscalar Giant Monopole Resonances. The model consists of a harmonic oscillator describing the collective excitation coupled with a nonlinear (Woods-Saxon) oscillator representing the motion of each nucléon. We are particulary interested in the dependence of relaxation on the energy of the system. We have found that in a rather broad region of parameter space, contrary to the common expectation, both Lyapunov exponent and relaxation time increase as a function of the total energy. We examine the conditions required for this effect to occur and demonstrate the key role of the dispersion relation of the nonlinear oscillator.

FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS

1994 ◽  
Vol 04 (04) ◽  
pp. 933-941 ◽  
Author(s):  
ANDREW L. KATZ ◽  
EARL H. DOWELL
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Stable And Unstable Manifolds ◽  
Detailed Explanation ◽  
Single Well ◽  
Unstable Manifolds ◽  
Parameter Values

The study of stable and unstable manifolds, and their intersections with each other, is a powerful technique for interpreting complex bifurcations of nonlinear systems. The escape phenomenon in the twin-well Duffing oscillator is one such bifurcation that is elucidated through the analysis of manifold intersections. In this paper, two escape scenarios in the twin-well Duffing oscillator are presented. In each scenario, the relevant manifold structures are examined for parameter values on either side of the escape bifurcation. Included is a description of the role of the hilltop saddle stable manifolds, which are known to separate the single well basins (should single well attractors exist). In each of the two bifurcation scenarios, it is shown through a detailed analysis of Poincaré maps that a homoclinic intersection of the manifolds of a specific period-3 saddle implies the destruction of the single well chaotic attractor. Although the Duffing oscillator is used to illustrate the ideas advanced here, it is thought that the approach will be useful for a variety of dynamical systems.

scholarly journals Response of Immunotherapy to Tumour-TICLs Interactions: A Travelling Wave Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Joseph Malinzi ◽  
Precious Sibanda ◽  
Hermane Mambili-Mamboundou
Keyword(s):  
Phase Space ◽  
Mathematical Modelling ◽  
Wave Speed ◽  
Interaction Model ◽  
Travelling Wave ◽  
Tumour Invasion ◽  
Wave Analysis ◽  
Stable And Unstable Manifolds ◽  
Minimum Wave Speed ◽  
Unstable Manifolds

There are several cancers for which effective treatment has not yet been identified. Mathematical modelling can nevertheless point out to clinicians tumour invasion properties that should be targeted to mitigate these cancers. We present a travelling wave analysis of a tumour-immune interaction model with immunotherapy. We use the geometric treatment of an apt-phase space to establish the intersection between stable and unstable manifolds. We calculate the minimum wave speed and numerical simulations are performed to support the analytical results.

Mechanics of Gels

1992 ◽  
Vol 271 ◽  
Author(s):  
George W. Scherer
Keyword(s):  
Thermal Expansion ◽  
Fluid Flow ◽  
Time Dependence ◽  
Supercritical Drying ◽  
Time Dependent ◽  
Dependent Behavior ◽  
Expansion Behavior ◽  
Pore Liquid ◽  
Applied Stresses

ABSTRACTThe response of wet gels to applied stresses is discussed, with emphasis on the role of flow of the pore liquid. Even when the network of the gel is purely elastic, the gel exhibits time-dependent behavior that resembles viscoelasticity, which results from fluid flow. The permeability of the gel can be determined from measurement of this time-dependence. Fluid flow also influences the thermal expansion behavior of gels, and can cause severe stresses to develop during supercritical drying, if the autoclave is heated too rapidly.

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