13,877 research outputs found

    A New Class of Two-dimensional Chaotic Maps with Closed Curve Fixed Points

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    This is the author accepted manuscript. The final version is available from World Scientific Publishing via the DOI in this recordThis paper constructs a new class of two-dimensional maps with closed curve fixed points. Firstly, the mathematical model of these maps is formulated by introducing a nonlinear function. Different types of fixed points which form a closed curve are shown by choosing proper parameters of the nonlinear function. The stabilities of these fixed points are studied to show that these fixed points are all non-hyperbolic. Then a computer search program is employed to explore the chaotic attractors in these maps, and several simple maps whose fixed points form different shapes of closed curves are presented. Complex dynamical behaviours of these maps are investigated by using the phase-basin portrait, Lyapunov exponents, and bifurcation diagrams.National Natural Science Foundation of ChinaNatural Science Foundation of Jiangsu Province of China5th 333 High-level Personnel Training Project of Jiangsu Province of ChinaExcellent Scientific and Technological Innovation Team of Jiangsu UniversityJiangsu Key Laboratory for Big Data of Psychology and Cognitive Scienc

    Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

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    Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems. While the former approach studies how regions of phase space are transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing periodic orbit around saddles. Both of these frameworks require computation with curves represented by millions of points-computing intersection points between these curves and area bounded by the segments of these curves-for quantifying the transport and escape rate. We present a theory for computing these intersection points and the area bounded between the segments of these curves based on a classification of the intersection points using equivalence class. We also present an alternate theory for curves with nontransverse intersections and a method to increase the density of points on the curves for locating the intersection points accurately.The numerical implementation of the theory presented herein is available as an open source software called Lober. We used this package to demonstrate the application of the theory to lobe dynamics that arises in fluid mechanics, and rate of escape from a potential well that arises in ship dynamics.Comment: 33 pages, 17 figure

    A global study of enhanced stretching and diffusion in chaotic tangles

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    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

    Topological Chaos in a Three-Dimensional Spherical Fluid Vortex

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    In chaotic deterministic systems, seemingly stochastic behavior is generated by relatively simple, though hidden, organizing rules and structures. Prominent among the tools used to characterize this complexity in 1D and 2D systems are techniques which exploit the topology of dynamically invariant structures. However, the path to extending many such topological techniques to three dimensions is filled with roadblocks that prevent their application to a wider variety of physical systems. Here, we overcome these roadblocks and successfully analyze a realistic model of 3D fluid advection, by extending the homotopic lobe dynamics (HLD) technique, previously developed for 2D area-preserving dynamics, to 3D volume-preserving dynamics. We start with numerically-generated finite-time chaotic-scattering data for particles entrained in a spherical fluid vortex, and use this data to build a symbolic representation of the dynamics. We then use this symbolic representation to explain and predict the self-similar fractal structure of the scattering data, to compute bounds on the topological entropy, a fundamental measure of mixing, and to discover two different mixing mechanisms, which stretch 2D material surfaces and 1D material curves in distinct ways.Comment: 14 pages, 11 figure

    Thirty Years of Turnstiles and Transport

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    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
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