7 research outputs found
Dynamics associated with a quasiperiodically forced Morse oscillator: Application to molecular dissociation
The dynamics associated with a quasiperiodically forced Morse oscillator is studied as a classical model for molecular dissociation under external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase-space barriers, allowing escape from bounded to unbounded motion. In contrast to the ubiquitous Poincaré map reduction of a periodically forced system, we derive a sequence of nonautonomous maps from the quasiperiodically forced system. We obtain a global picture of the dynamics, i.e., of transport in phase space, using a sequence of time-dependent two-dimensional lobe structures derived from the invariant homoclinic tangle of a persisting invariant saddle-type torus in a Poincaré section of an associated autonomous system phase space. Transport is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, and this provides the framework for studying basic features of molecular dissociation in the context of classical phase-space trajectories. We obtain a precise criterion for discerning between bounded and unbounded motion in the context of the forced problem. We identify and measure analytically the flux associated with the transition between bounded and unbounded motion, and study dissociation rates for a variety of initial phase-space ensembles, such as an even or weighted distribution of points in phase space, or a distribution on a particular level set of the unperturbed Hamiltonian (corresponding to a quantum state). A double-phase-slice sampling method allows exact numerical computation of dissociation rates. We compare single- and two-frequency forcing. Infinite-time average flux is maximal in a particular single-frequency limit; however, lobe penetration of the level sets of the unperturbed Hamiltonian can be maximal in the two-frequency case. The variation of lobe areas in the two-frequency problem gives one added freedom to enhance or diminish aspects of phase-space transport on finite time scales for a fixed infinite-time average flux, and for both types of forcing the geometry of lobes is relevant. The chaotic nature of the dynamics is understood in terms of a traveling horseshoe map sequence
Statistical relaxation under nonturbulent chaotic flows: Non-Gaussian high-stretch tails of finite-time Lyapunov exponent distributions
We observe that high-stretch tails of finite-time Lyapunov exponent distributions associated with interfaces evolving under a class of nonturbulent chaotic flows can range from essentially Gaussian tails to nearly exponential tails, and show that the non-Gaussian deviations can have a significant effect on interfacial evolution. This observation motivates new insight into stretch processes under chaotic flows
Transport, stretching, and mixing in classes of chaotic tangles
We use global stable and unstable manifolds of invariant hyperbolic sets as templates
for studying the dynamics within classes of homoclinic and heteroclinic chaotic tangles,
focusing on transport, stretching, and mixing within these tangles. These templates are
exploited in the context of lobes in phase space mapping within invariant lobe structures
formed out of the intersecting global stable and unstable manifolds. Our interest lies
in: (a) extending the templates and their applications to fundamentally larger classes of
dynamical systems, (b) expanding the description of dynamics offered by the templates,
and (c) applying the templates to the study of various nonlinear physical phenomena,
such as stirring and mixing under chaotically advecting fluids and molecular dissociation
under external electromagnetic forcing. These and other nonlinear physical phenomena
are intimately connected to the underlying chaotic dynamics, and describing these processes
encourages study of finite-time, or transient, phenomena as well as asymptotics,
the former being much more virgin territory from a dynamical systems perspective. Under
the rubric of themes (a)-(c) we offer five studies.
(i) One of the canonical classes of dynamical systems in which these templates have
been exploited is defined by 2D time-periodic vector fields, where the analysis reduces
to a 2D Poincaré map. In this instance, one is well-equipped with basic elements of
dynamical systems theory associated with 2D maps, such as the Smale horseshoe map
paradigm, KAM-tori, hyperbolic fixed points and their global stable and unstable manifolds
that define the tangle boundaries, classical Melnikov theory, and so on. Our first
study performs a systematic extension of the dynamical system constructs associated
with 2D time-periodic vector fields to apply to 2D vector fields with more complicated
time dependences. In particular, we focus on 2D vector fields with quasiperiodic, or
multiple-frequency, time dependence. Any extension past the time-periodic case requires
the fundamental generalization from 2D maps to sequences of 2D nonautonomous
maps. To large extent the constructs associated with 2D Poincaré maps are found to
be robust under this generalization. For example, the Smale horseshoe map generalizes
to a traveling horseshoe map sequence, hyperbolic fixed points generalize to points that
live on invariant normally hyperbolic tori, and invariant 2D chaotic tangles generalize to
sequences of 2D chaotic tangles derived from an invariant tangle embedded in a higher-dimensional
phase space. It is within the setting of 2D lobes mapping within a sequence
of 2D lobe structures that one has a template for systematic study of the dynamics
generated by multiple-frequency vector fields. Dynamical systems tools with which to
study these systems include: (i) a generalized Melnikov theory that offers an approximate
analytical measure of stable and unstable manifold separation in the tangles, the
basis for a variety of analytical studies, and (ii) a double phase slice sampling method
that allows for numerical computation of precise 2D slices of the higher-dimensional invariant
chaotic tangles, the basis for numerical work. The Melnikov function defines
relative scaling functions which give an analytical measure of the relative importance of
each frequency on manifold separation. With the template and tools in hand, we study
multiple-frequency dynamics and compare with single-frequency dynamics. We recast
lobe dynamics under a hi-infinite sequence of nonautonomous maps in closed form by
exploiting underlying periodicity properties of the vector field, and present numerical
simulations of sequences of chaotic tangles and lobe dynamics within these tangles. In
contrast to lobes of equal area mapping within a fixed 2D lobe structure found under
single-frequency forcing, we find lobes of varying areas mapping within a sequence of
lobe structures that are distorting and breathing from one time sample to the next, affording
greater freedom in the nature of the dynamics. Our primary focus in this new
setting is on phase space transport (we consider stretching and mixing in other contexts
in later studies). The non-integrable motion in chaotic tangles allows for transport between
various regions of phase space, in particular, between regions corresponding to
qualitatively different types of motion, such as bounded and unbounded motion. This
transport is intimately connected to basic physical processes, such as the fluid mixing
and molecular dissociation processes. Transport theory refers to the enterprise where
one uses a combination of invariant manifold theory, Melnikov theory, numerical simulation
and/or approximate models such as Markov models, to partition phase space into
regions of qualitatively different behavior (such as bounded and unbounded motion),
establish complete and partial barriers between the regions, identify the turnstile lobes
that are the gateways for transport across partial barriers, and then study in the context
of lobe dynamics such phase space transport issues as flux and escape rates from a
particular region. The formal construction of a transport theory for multiple-frequency
vector fields is more involved than in the single-frequency case, as a consequence of more
complicated manifold geometry. This geometry is uncovered and explored, however, via
theorems and numerical studies based on Melnikov theory. We then partition phase
space and define turnstiles in the higher-dimensional autonomous setting, and from this
obtain the sequence of partitions and turnstiles in the 2D nonautonomous setting. A
main new feature of transport is its manifestation in the context of a sequence of time-dependent
regions, and we argue this is consistent with a Lagrangian viewpoint. We
then perform a detailed study of such transport properties as flux, lobe geometry, and
lobe content. In contrast to the single-frequency case, where a single flux suffices, in the
multiple-frequency case a variety of fluxes are allowed, such as different types of instantaneous,
finite-time average, and infinite-time average flux. We find for certain classes
of multiple-frequency forcing that infinite-time average flux is maximal in a particular
single-frequency limit, but that the spatial variation of lobe areas found in multiple-frequency
systems affords greater freedom to enhance or diminish finite-time transport
quantities. We illustrate our study with a quasiperiodically oscillating vortical flow that
gives rise to chaotic fluid trajectories and a quasiperiodically forced Duffing oscillator.
We explain how the analysis generalizes to vector fields with more complicated time
dependences than quasiperiodic.
(ii) Besides the destruction of phase space barriers, allowing for phase space transport,
other essential features of the dynamics in chaotic tangles include greatly enhanced
stretching and mixing. Our second study returns to 2D time-periodic vector fields and
uses invariant manifolds as templates for a global study of stretching and mixing in
chaotic tangles. The analysis here thus complements the one of transport via invariant
manifolds, and can essentially be viewed as a generalization of the horseshoe map construction
to apply to entire material interfaces inside the tangles. Given the dominant
role of the unstable manifold in chaotic tangles, we study the stretching of a material interface
originating on a segment of the unstable manifold associated with a turnstile lobe.
We construct a symbolic dynamics formalism that describes the evolution of the entire
material curve, which is the basis for a global understanding of the stretch processes in
chaotic tangles, such as the topology of stretching, the mechanisms for good stretching,
and the statistics of stretching. A central interest will be in understanding the stretch
profile of the material interface, which is the graph of finite-time stretch experienced as
a function of location on the interface. In a near-integrable setting (meaning we add a
perturbation to the vector field of an originally integrable system) we argue how the perturbed
stretch profile can be understood in terms of a corresponding unperturbed stretch
profile approximately repeating itself on smaller and smaller scales, as described by the
symbolic dynamics. The basic interest is in how the non-uniformity in the unperturbed
stretch profile can approximately carry through to the non-uniformity in the perturbed
stretch profile, and this non-uniformity can play a basic role in mixing properties and
stretch statistics. After the stretch analysis we then add to the deterministic flows a small
stochastic component, corresponding for example to molecular diffusion (with small diffusion
coefficient D) in a fluid flow, and study the diffusion of passive scalars across
material interfaces inside the chaotic tangles. For sufficiently thin diffusion zones, the
diffusion of passive scalars across interfaces can be treated as a one-dimensional process,
and diffusion rates across interfaces are directly related to the stretch history of the interface.
Our understanding of stretching thus directly translates into an understanding
of mixing. However, a notable exception to the thin diffusion zone approximation occurs
when an interface folds on top of itself so that neighboring diffusion zones overlap. We
present an analysis which takes into account the overlap of neighboring diffusion zones,
capturing a saturation effect in the diffusion process relevant to efficiency of mixing. We
illustrate the stretching and mixing study in the context of two oscillating vortex pair
flows, one corresponding to an open heteroclinic tangle, the other to a closed homoclinic
tangle. Though we focus here on single-frequency systems, from the previous study the
extensions to multiple-frequency systems should be clear.
(iii) We then study stretching from a different perspective, focusing on rates of
strain experienced by infinitesimal line elements as they evolve under near-integrable
chaotic flows associated with 2D time-periodic velocity fields. We introduce the notion
of irreversible rate of strain responsible for net stretch, study the role of hyperbolic fixed
points as engines for good irreversible straining, and observe the role of turnstiles as
mechanisms for enhancing straining efficiency via re-orientation of line elements and
transport of line elements to regions of superior straining.
(iv) The remaining two studies can be viewed as applications of the material developed
in the previous studies, although both applications develop new theory and/or new
ideas as well. The first application studies the dynamics associated with a quasi-periodically
forced Morse oscillator as a classical model for molecular dissociation under
external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase
space barriers, allowing escape from bounded to unbounded motion, and we study this
transition in the context of our quasiperiodic theory, comparing with single-frequency
forcing. New and interesting features of this application beyond the subject matter of
the previous quasiperiodic study includes that the relevant fixed point of the unforced
system is non-hyperbolic and at infinity, and the study of additional transport issues,
such as escape (implying dissociation) from a particular level set of the unforced Hamiltonian
system corresponding to a quantum state. We find for example that though
infinite-time average flux can be maximal in a single-frequency limit, escape from a level
set, or equivalently lobe penetration, can be maximal in the multiple-frequency case.
(v) The second application studies statistical relaxation of distributions of finite-time
Lyapunov exponents associated with interfaces evolving within the chaotic tangles
of 2D time-periodic vector fields. Whereas recent studies claim or give evidence that
distributions of finite-time Lyapunov exponents are essentially Gaussian, our previous
analysis of stretching via the symbolic dynamics construction shows the wide variety
of stretch processes and stretch scales involved in the tangle, motivating our further
study of stretch statistics. In particular, we focus on the high-stretch tails of finite-time
Lyapunov exponents, which have relevance in incompressible flows to the mixing
properties and multifractal characteristics of passive scalars and vectors in the limit of
small spatial scales. Previous studies of stretch distributions consider a fixed number of
points, thus lacking adequate resolution to study these tails. Instead, we use a dynamic
point insertion scheme to maintain adequate interfacial covering, entailing extremely
good resolution at high-stretch tails. These tails show a great range in behavior, varying
from essentially Gaussian to nearly exponential, and these non-Gaussian deviations can
have a significant effect on interfacial stretching, one that persists asymptotically. These
non-Gaussian deviations can be associated with very small probabilities, thus indicating
the need for highly-resolved numerical studies of stretch statistics. We explain the nearly
exponential tail in a particular limiting regime corresponding to highly non-uniform
stretch profiles, and explore how the full statistics might be captured by elementary
models for the stretch processes.</p