1,402 research outputs found
Thirty Years of Turnstiles and Transport
To characterize transport in a deterministic dynamical system is to compute
exit time distributions from regions or transition time distributions between
regions in phase space. This paper surveys the considerable progress on this
problem over the past thirty years. Primary measures of transport for
volume-preserving maps include the exiting and incoming fluxes to a region. For
area-preserving maps, transport is impeded by curves formed from invariant
manifolds that form partial barriers, e.g., stable and unstable manifolds
bounding a resonance zone or cantori, the remnants of destroyed invariant tori.
When the map is exact volume preserving, a Lagrangian differential form can be
used to reduce the computation of fluxes to finding a difference between the
action of certain key orbits, such as homoclinic orbits to a saddle or to a
cantorus. Given a partition of phase space into regions bounded by partial
barriers, a Markov tree model of transport explains key observations, such as
the algebraic decay of exit and recurrence distributions.Comment: Updated and corrected versio
Phase space geometry and reaction dynamics near index two saddles
We study the phase space geometry associated with index 2 saddles of a
potential energy surface and its influence on reaction dynamics for
degree-of-freedom (DoF) Hamiltonian systems. For index 1 saddles of potential
energy surfaces (the case of classical transition state theory), the existence
of a normally hyperbolic invariant manifold (NHIM) of saddle stability type has
been shown, where the NHIM serves as the "anchor" for the construction of
dividing surfaces having the no-recrossing property and minimal flux. For the
index 1 saddle case the stable and unstable manifolds of the NHIM are
co-dimension one in the energy surface, and act as conduits for reacting
trajectories in phase space. The situation for index 2 saddles is quite
different. We show that NHIMs with their stable and unstable manifolds still
exist, but that these manifolds by themselves lack sufficient dimension to act
as barriers in the energy surface. Rather, there are different types of
invariant manifolds, containing the NHIM and its stable and unstable manifolds,
that act as co-dimension one barriers in the energy surface. These barriers
divide the energy surface in the vicinity of the index 2 saddle into regions of
qualitatively different trajectories exhibiting a wider variety of dynamical
behavior than for the case of index 1 saddles. In particular, we can identify a
class of trajectories, which we refer to as "roaming trajectories", which are
not associated with reaction along the classical minimum energy path (MEP). We
illustrate the significance of our analysis of the index 2 saddle for reaction
dynamics with two examples.Comment: 43 pages, 4 figure
Geometrical Models of the Phase Space Structures Governing Reaction Dynamics
Hamiltonian dynamical systems possessing equilibria of stability type display \emph{reaction-type
dynamics} for energies close to the energy of such equilibria; entrance and
exit from certain regions of the phase space is only possible via narrow
\emph{bottlenecks} created by the influence of the equilibrium points. In this
paper we provide a thorough pedagogical description of the phase space
structures that are responsible for controlling transport in these problems. Of
central importance is the existence of a \emph{Normally Hyperbolic Invariant
Manifold (NHIM)}, whose \emph{stable and unstable manifolds} have sufficient
dimensionality to act as separatrices, partitioning energy surfaces into
regions of qualitatively distinct behavior. This NHIM forms the natural
(dynamical) equator of a (spherical) \emph{dividing surface} which locally
divides an energy surface into two components (`reactants' and `products'), one
on either side of the bottleneck. This dividing surface has all the desired
properties sought for in \emph{transition state theory} where reaction rates
are computed from the flux through a dividing surface. In fact, the dividing
surface that we construct is crossed exactly once by reactive trajectories, and
not crossed by nonreactive trajectories, and related to these properties,
minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space
structures contained in it for 2-degree-of-freedom (DoF) systems in the
threedimensional space , and two schematic models which capture many of
the essential features of the dynamics for -DoF systems. In addition, we
elucidate the structure of the NHIM.Comment: 44 pages, 38 figures, PDFLaTe
Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
In this tutorial, we discuss self-excited and hidden attractors for systems
of differential equations. We considered the example of a Lorenz-like system
derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to
demonstrate the analysis of self-excited and hidden attractors and their
characteristics. We applied the fishing principle to demonstrate the existence
of a homoclinic orbit, proved the dissipativity and completeness of the system,
and found absorbing and positively invariant sets. We have shown that this
system has a self-excited attractor and a hidden attractor for certain
parameters. The upper estimates of the Lyapunov dimension of self-excited and
hidden attractors were obtained analytically.Comment: submitted to EP
Numerical investigation of chaotic dynamics in multidimensional transition states
Many chemical reactions can be described as the crossing of an energetic barrier. This process is mediated by an invariant object in phase space. One can construct a normally hyperbolic invariant manifold (NHIM) of the reactive dynamical system which is an invariant sphere that can be considered as the geometric representation of the transition state itself. The NHIM has invariant cylinders (reaction channels) attached to it. This invariant geometric structure survives as long as the invariant sphere is normally hyperbolic. We applied this theory to the hydrogen exchange reaction in three degrees of freedom in order to figure out the reason of the transition state theory (TST) failure. Energies high above the reaction threshold, the dynamics within the transition state becomes partially chaotic. We have found that the invariant sphere first ceases to be normally hyperbolic at fairly low energies. Surprisingly normal hyperbolicity is then restored and the invariant sphere remains normally hyperbolic even at very high energies. This observation shows two different energy values for the breakdown of the TST and the breakdown of the NHIM.
This leads to seek another phase space object that is related to the breakdown of the TST. Using theory of the dividing surface including reactive islands (RIs), we can investigate such an object. We found out that the first nonreactive trajectory has been found at the same energy values for both collinear and full systems, and coincides with the first bifurcation of periodic orbit dividing surface (PODS) at the collinear configuration. The bifurcation creates the unstable periodic orbit (UPO). Indeed, the new PODS (UPO) is the reason for the TST failure. The manifolds (stable and centre-stable) of the UPO clarify these expectations by intersecting the dividing surface at the boundary of the reactive island (on the collinear and the three (full) systems, respectively)
Investigation and visualization of the stability boundary for stressed power systems
Present interconnected power systems are being pushed to their limits due to heavier loading of the transmission network and delay in facility construction. The resultant vulnerability in withstanding system disturbances requires a more accurate understanding of system stability behavior. This dissertation presents the use of real normal form of vector fields, a comparatively new analysis tool in the area of power systems, combined with the use of XGobi, an effective graphic package for multi-dimensional visualization, to investigate and visualize the stability boundary of the stressed system. It also depicts the structural characteristics for the stressed power system corresponding to specific fault scenarios by computing the participation factors and the nonlinear indices along the actual fault trajectory which is obtained from time simulation of the equations governing system dynamics. The objective of this research is to analyze and explain the nonlinear phenomena in stressed power systems and characterize the stability boundary of the power system when subjected to large disturbances. The structural information provided by the method of normal forms will also be utilized in explaining the mechanism of instability near the stability boundary and will be used to determine the critical interactions involved;The main idea of this dissertation is to characterize the stability boundary of the stressed power system by obtaining the nonlinear structural information through the visualization of the stability boundary in multiple dimensions, and by utilizing the analytical measures of nonlinear interaction indices and nonlinear participation factors obtained using the method of normal forms;The proposed approach has been tested on the IEEE 4-generator system and 11-generator system. The stability boundary of the system is approximated by a second order stable manifold of the controlling unstable equilibrium point. The stable manifold is constructed by spanning all stable directions. The effective graphic tool XGobi is used to visualize the approximated stability boundary in all dimensions of the system, which helps to obtain a global structural information of the system. The shape and curvature of the stability boundary are detected. An extended approach to deal with large sized power systems is studied, which provide an effective method to study large sized power systems. The computation of linear and nonlinear participation factors, together with the nonlinear indices of the fault trajectory obtained by the time simulation provides the basis to study the structural characteristics of the system. The structural information helps the tuning of control action, which is valuable for maintaining the stability of practical power systems
Numerical studies to detect chaotic motion in the full planar averaged three-body problem
In this paper, the author deals with a well-known problem of Celestial Mechanics, namely the three-body problem. A numerical analysis has been done in order to prove existence of chaotic motions of the full-averaged problem in particular configurations. Full because all the three bodies have non-negligible masses and averaged because the Hamiltonian describing the system has been averaged with respect to a fast angle. A reduction of degrees of freedom and of the phase-space is performed in order to apply the notion of covering relations and symbolic dynamics
Point-set manifold processing for computational mechanics: thin shells, reduced order modeling, cell motility and molecular conformations
In many applications, one would like to perform calculations on smooth manifolds of dimension d embedded in a high-dimensional space of dimension D. Often, a continuous description of such manifold is not known, and instead it is sampled by a set of scattered points in high dimensions. This poses a serious challenge. In this thesis, we approximate the point-set manifold as an overlapping set of smooth parametric descriptions, whose geometric structure is revealed by statistical learning methods, and then parametrized by meshfree methods. This approach avoids any global parameterization, and hence is applicable to manifolds of any genus and complex geometry. It combines four ingredients: (1) partitioning of the point set into subregions of trivial topology, (2) the automatic detection of the local geometric structure of the manifold by nonlinear dimensionality reduction techniques, (3) the local parameterization of the manifold using smooth meshfree (here local maximum-entropy) approximants, and (4) patching together the local representations by means of a partition of unity.
In this thesis we show the generality, flexibility, and accuracy of the method in four different problems. First, we exercise it in the context of Kirchhoff-Love thin shells, (d=2, D=3). We test our methodology against classical linear and non linear benchmarks in thin-shell analysis, and highlight its ability to handle point-set surfaces of complex topology and geometry. We then tackle problems of much higher dimensionality. We perform reduced order modeling in the context of finite deformation elastodynamics, considering a nonlinear reduced configuration space, in contrast with classical linear approaches based on Principal Component Analysis (d=2, D=10000's). We further quantitatively unveil the geometric structure of the motility strategy of a family of micro-organisms called Euglenids from experimental videos (d=1, D~30000's). Finally, in the context of enhanced sampling in molecular dynamics, we automatically construct collective variables for the molecular conformational dynamics (d=1...6, D~30,1000's)
Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening
This work introduces a number of algebraic topology approaches, such as
multicomponent persistent homology, multi-level persistent homology and
electrostatic persistence for the representation, characterization, and
description of small molecules and biomolecular complexes. Multicomponent
persistent homology retains critical chemical and biological information during
the topological simplification of biomolecular geometric complexity.
Multi-level persistent homology enables a tailored topological description of
inter- and/or intra-molecular interactions of interest. Electrostatic
persistence incorporates partial charge information into topological
invariants. These topological methods are paired with Wasserstein distance to
characterize similarities between molecules and are further integrated with a
variety of machine learning algorithms, including k-nearest neighbors, ensemble
of trees, and deep convolutional neural networks, to manifest their descriptive
and predictive powers for chemical and biological problems. Extensive numerical
experiments involving more than 4,000 protein-ligand complexes from the PDBBind
database and near 100,000 ligands and decoys in the DUD database are performed
to test respectively the scoring power and the virtual screening power of the
proposed topological approaches. It is demonstrated that the present approaches
outperform the modern machine learning based methods in protein-ligand binding
affinity predictions and ligand-decoy discrimination
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