[ { "id": "authors:kkjzx-pvc73", "collection": "authors", "collection_id": "kkjzx-pvc73", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BEIprl93", "type": "article", "title": "Statistical relaxation under nonturbulent chaotic flows: Non-Gaussian high-stretch tails of finite-time Lyapunov exponent distributions", "author": [ { "family_name": "Beigie", "given_name": "Darin", "clpid": "Beigie-D" }, { "family_name": "Leonard", "given_name": "Anthony", "clpid": "Leonard-A" }, { "family_name": "Wiggins", "given_name": "Stephen", "clpid": "Wiggins-S" } ], "abstract": "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.", "doi": "10.1103/PhysRevLett.70.275", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1993-01-18", "series_number": "3", "volume": "70", "issue": "3", "pages": "275-278" }, { "id": "authors:wnasx-1dy74", "collection": "authors", "collection_id": "wnasx-1dy74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BEIpra92", "type": "article", "title": "Dynamics associated with a quasiperiodically forced Morse oscillator: Application to molecular dissociation", "author": [ { "family_name": "Beigie", "given_name": "Darin", "clpid": "Beigie-D" }, { "family_name": "Wiggins", "given_name": "Stephen", "clpid": "Wiggins-S" } ], "abstract": "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 Poincar\u00e9 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 Poincar\u00e9 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.", "doi": "10.1103/PhysRevA.45.4803", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1992-04-01", "series_number": "7", "volume": "45", "issue": "7", "pages": "4803-4829" }, { "id": "authors:9wche-sra06", "collection": "authors", "collection_id": "9wche-sra06", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BEInonlin91", "type": "article", "title": "Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems", "author": [ { "family_name": "Beigie", "given_name": "Darren", "clpid": "Beigie-D" }, { "family_name": "Leonard", "given_name": "Anthony", "clpid": "Leonard-A" }, { "family_name": "Wiggins", "given_name": "Stephen", "clpid": "Wiggins-S" } ], "abstract": "The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles.", "doi": "10.1088/0951-7715/4/3/008", "issn": "0951-7715", "publisher": "Nonlinearity", "publication": "Nonlinearity", "publication_date": "1991-08-01", "series_number": "3", "volume": "4", "issue": "3", "pages": "775-819" }, { "id": "authors:yw272-vph18", "collection": "authors", "collection_id": "yw272-vph18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BEIpofa91", "type": "article", "title": "A global study of enhanced stretching and diffusion in chaotic tangles", "author": [ { "family_name": "Beigle", "given_name": "Darin", "clpid": "Beigie-D" }, { "family_name": "Leonard", "given_name": "Anthony", "clpid": "Leonard-A" }, { "family_name": "Wiggins", "given_name": "Stephen", "clpid": "Wiggins-S" } ], "abstract": "A global, finite-time study is made of stretching and diffusion in a class of chaotic tangles associated with fluids described by periodically forced two-dimensional dynamical systems. Invariant lobe structures formed by intersecting global stable and unstable manifolds of persisting invariant hyperbolic sets provide the geometrical framework for studying stretching of interfaces and diffusion of passive scalars across these interfaces. In particular, the present study focuses on the material curve that initially lies on the unstable manifold segment of the boundary of the entraining turnstile lobe.A knowledge of the stretch profile of a corresponding curve that evolves according to the unperturbed flow, coupled with an appreciation of a symbolic dynamics that applies to the entire original material curve in the perturbed flow, provides the framework for understanding the mechanism for, and topology of, enhanced stretching in chaotic tangles. Secondary intersection points (SIP's) of the stable and unstable manifolds are particularly relevant to the topology, and the perturbed stretch profile is understood in terms of the unperturbed stretch profile approximately repeating itself on smaller and smaller scales. For sufficiently thin diffusion zones, 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.An understanding of interface stretching thus directly translates to an understanding of diffusion across interfaces. 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. An analysis which takes into account the overlap of nearest neighbor diffusion zones is presented, which is sufficient to capture new phenomena relevant to efficiency of mixing. The analysis adds to the concentration profile a saturation term that depends on the distance between neighboring segments of the interface. Efficiency of diffusion thus depends not only on efficiency of stretching along the interface, but on how this stretching is distributed relative to the distance between neighboring segments of the interface.", "doi": "10.1063/1.858084", "issn": "0899-8213", "publisher": "American Institute of Physics", "publication": "Physics of Fluids A", "publication_date": "1991-05", "series_number": "5", "volume": "3", "issue": "5", "pages": "1039-1050" } ]