We use global stable and unstable manifolds of invariant hyperbolic sets as templatesfor studying the dynamics within classes of homoclinic and heteroclinic chaotic tangles,focusing on transport, stretching, and mixing within these tangles. These templates areexploited in the context of lobes in phase space mapping within invariant lobe structuresformed out of the intersecting global stable and unstable manifolds. Our interest liesin: (a) extending the templates and their applications to fundamentally larger classes ofdynamical 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 dissociationunder external electromagnetic forcing. These and other nonlinear physical phenomenaare intimately connected to the underlying chaotic dynamics, and describing these processesencourages study of finite-time, or transient, phenomena as well as asymptotics,the former being much more virgin territory from a dynamical systems perspective. Underthe rubric of themes (a)-(c) we offer five studies.
(i) One of the canonical classes of dynamical systems in which these templates havebeen exploited is defined by 2D time-periodic vector fields, where the analysis reducesto a 2D Poincaré map. In this instance, one is well-equipped with basic elements ofdynamical systems theory associated with 2D maps, such as the Smale horseshoe mapparadigm, KAM-tori, hyperbolic fixed points and their global stable and unstable manifoldsthat define the tangle boundaries, classical Melnikov theory, and so on. Our firststudy performs a systematic extension of the dynamical system constructs associatedwith 2D time-periodic vector fields to apply to 2D vector fields with more complicatedtime dependences. In particular, we focus on 2D vector fields with quasiperiodic, ormultiple-frequency, time dependence. Any extension past the time-periodic case requiresthe fundamental generalization from 2D maps to sequences of 2D nonautonomousmaps. To large extent the constructs associated with 2D Poincaré maps are found tobe robust under this generalization. For example, the Smale horseshoe map generalizesto a traveling horseshoe map sequence, hyperbolic fixed points generalize to points thatlive on invariant normally hyperbolic tori, and invariant 2D chaotic tangles generalize tosequences of 2D chaotic tangles derived from an invariant tangle embedded in a higher-dimensionalphase space. It is within the setting of 2D lobes mapping within a sequenceof 2D lobe structures that one has a template for systematic study of the dynamicsgenerated by multiple-frequency vector fields. Dynamical systems tools with which tostudy these systems include: (i) a generalized Melnikov theory that offers an approximateanalytical measure of stable and unstable manifold separation in the tangles, thebasis for a variety of analytical studies, and (ii) a double phase slice sampling methodthat allows for numerical computation of precise 2D slices of the higher-dimensional invariantchaotic tangles, the basis for numerical work. The Melnikov function definesrelative scaling functions which give an analytical measure of the relative importance ofeach frequency on manifold separation. With the template and tools in hand, we studymultiple-frequency dynamics and compare with single-frequency dynamics. We recastlobe dynamics under a hi-infinite sequence of nonautonomous maps in closed form byexploiting underlying periodicity properties of the vector field, and present numericalsimulations of sequences of chaotic tangles and lobe dynamics within these tangles. Incontrast to lobes of equal area mapping within a fixed 2D lobe structure found undersingle-frequency forcing, we find lobes of varying areas mapping within a sequence oflobe structures that are distorting and breathing from one time sample to the next, affordinggreater freedom in the nature of the dynamics. Our primary focus in this newsetting is on phase space transport (we consider stretching and mixing in other contextsin later studies). The non-integrable motion in chaotic tangles allows for transport betweenvarious regions of phase space, in particular, between regions corresponding toqualitatively different types of motion, such as bounded and unbounded motion. Thistransport is intimately connected to basic physical processes, such as the fluid mixingand molecular dissociation processes. Transport theory refers to the enterprise whereone uses a combination of invariant manifold theory, Melnikov theory, numerical simulationand/or approximate models such as Markov models, to partition phase space intoregions of qualitatively different behavior (such as bounded and unbounded motion),establish complete and partial barriers between the regions, identify the turnstile lobesthat are the gateways for transport across partial barriers, and then study in the contextof lobe dynamics such phase space transport issues as flux and escape rates from aparticular region. The formal construction of a transport theory for multiple-frequencyvector fields is more involved than in the single-frequency case, as a consequence of morecomplicated manifold geometry. This geometry is uncovered and explored, however, viatheorems and numerical studies based on Melnikov theory. We then partition phasespace and define turnstiles in the higher-dimensional autonomous setting, and from thisobtain the sequence of partitions and turnstiles in the 2D nonautonomous setting. Amain new feature of transport is its manifestation in the context of a sequence of time-dependentregions, and we argue this is consistent with a Lagrangian viewpoint. Wethen perform a detailed study of such transport properties as flux, lobe geometry, andlobe content. In contrast to the single-frequency case, where a single flux suffices, in themultiple-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 classesof multiple-frequency forcing that infinite-time average flux is maximal in a particularsingle-frequency limit, but that the spatial variation of lobe areas found in multiple-frequencysystems affords greater freedom to enhance or diminish finite-time transportquantities. We illustrate our study with a quasiperiodically oscillating vortical flow thatgives rise to chaotic fluid trajectories and a quasiperiodically forced Duffing oscillator.We explain how the analysis generalizes to vector fields with more complicated timedependences 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 enhancedstretching and mixing. Our second study returns to 2D time-periodic vector fields anduses invariant manifolds as templates for a global study of stretching and mixing inchaotic tangles. The analysis here thus complements the one of transport via invariantmanifolds, and can essentially be viewed as a generalization of the horseshoe map constructionto apply to entire material interfaces inside the tangles. Given the dominantrole of the unstable manifold in chaotic tangles, we study the stretching of a material interfaceoriginating on a segment of the unstable manifold associated with a turnstile lobe.We construct a symbolic dynamics formalism that describes the evolution of the entirematerial curve, which is the basis for a global understanding of the stretch processes inchaotic 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 stretchprofile of the material interface, which is the graph of finite-time stretch experienced asa function of location on the interface. In a near-integrable setting (meaning we add aperturbation to the vector field of an originally integrable system) we argue how the perturbedstretch profile can be understood in terms of a corresponding unperturbed stretchprofile approximately repeating itself on smaller and smaller scales, as described by thesymbolic dynamics. The basic interest is in how the non-uniformity in the unperturbedstretch profile can approximately carry through to the non-uniformity in the perturbedstretch profile, and this non-uniformity can play a basic role in mixing properties andstretch statistics. After the stretch analysis we then add to the deterministic flows a smallstochastic component, corresponding for example to molecular diffusion (with small diffusioncoefficient D) in a fluid flow, and study the diffusion of passive scalars acrossmaterial interfaces inside the chaotic tangles. For sufficiently thin diffusion zones, thediffusion 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 understandingof mixing. However, a notable exception to the thin diffusion zone approximation occurswhen an interface folds on top of itself so that neighboring diffusion zones overlap. Wepresent 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. Weillustrate the stretching and mixing study in the context of two oscillating vortex pairflows, one corresponding to an open heteroclinic tangle, the other to a closed homoclinictangle. Though we focus here on single-frequency systems, from the previous study theextensions to multiple-frequency systems should be clear.
(iii) We then study stretching from a different perspective, focusing on rates ofstrain experienced by infinitesimal line elements as they evolve under near-integrablechaotic flows associated with 2D time-periodic velocity fields. We introduce the notionof irreversible rate of strain responsible for net stretch, study the role of hyperbolic fixedpoints as engines for good irreversible straining, and observe the role of turnstiles asmechanisms for enhancing straining efficiency via re-orientation of line elements andtransport of line elements to regions of superior straining.
(iv) The remaining two studies can be viewed as applications of the material developedin the previous studies, although both applications develop new theory and/or newideas as well. The first application studies the dynamics associated with a quasi-periodicallyforced Morse oscillator as a classical model for molecular dissociation underexternal quasiperiodic electromagnetic forcing. The forcing entails destruction of phasespace barriers, allowing escape from bounded to unbounded motion, and we study thistransition in the context of our quasiperiodic theory, comparing with single-frequencyforcing. New and interesting features of this application beyond the subject matter ofthe previous quasiperiodic study includes that the relevant fixed point of the unforcedsystem 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 Hamiltoniansystem corresponding to a quantum state. We find for example that thoughinfinite-time average flux can be maximal in a single-frequency limit, escape from a levelset, or equivalently lobe penetration, can be maximal in the multiple-frequency case.
(v) The second application studies statistical relaxation of distributions of finite-timeLyapunov exponents associated with interfaces evolving within the chaotic tanglesof 2D time-periodic vector fields. Whereas recent studies claim or give evidence thatdistributions of finite-time Lyapunov exponents are essentially Gaussian, our previousanalysis of stretching via the symbolic dynamics construction shows the wide varietyof stretch processes and stretch scales involved in the tangle, motivating our furtherstudy of stretch statistics. In particular, we focus on the high-stretch tails of finite-timeLyapunov exponents, which have relevance in incompressible flows to the mixingproperties and multifractal characteristics of passive scalars and vectors in the limit ofsmall spatial scales. Previous studies of stretch distributions consider a fixed number ofpoints, thus lacking adequate resolution to study these tails. Instead, we use a dynamicpoint insertion scheme to maintain adequate interfacial covering, entailing extremelygood resolution at high-stretch tails. These tails show a great range in behavior, varyingfrom essentially Gaussian to nearly exponential, and these non-Gaussian deviations canhave a significant effect on interfacial stretching, one that persists asymptotically. Thesenon-Gaussian deviations can be associated with very small probabilities, thus indicatingthe need for highly-resolved numerical studies of stretch statistics. We explain the nearlyexponential tail in a particular limiting regime corresponding to highly non-uniformstretch profiles, and explore how the full statistics might be captured by elementarymodels for the stretch processes.
