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 Title
 A Local Regression Approach to Computing the Cauchy Green Strain Tensor.
 Creator
 Kepley, Shane, Kalies, William D., Graduate College
 Abstract/Description

The Cauchy Green strain tensor provides an effective tool for understanding unsteady flows. In particular, the dominant eigenvalue of this tensor has been seen to be a reliable estimator of the finite time Lyapunov exponent. We propose a new method for computing the CG strain tensor using a local quadratic regression LOESS technique. We compare this LOESS method with several classical methods using closed form flows, noisy flows, and simulated time series. In each case, the CG strain tensor...
Show moreThe Cauchy Green strain tensor provides an effective tool for understanding unsteady flows. In particular, the dominant eigenvalue of this tensor has been seen to be a reliable estimator of the finite time Lyapunov exponent. We propose a new method for computing the CG strain tensor using a local quadratic regression LOESS technique. We compare this LOESS method with several classical methods using closed form flows, noisy flows, and simulated time series. In each case, the CG strain tensor produced by the LOESS method is remarkably accurate and robust compared to classical methods.
Show less  Date Issued
 2015
 PURL
 http://purl.flvc.org/fau/fd/FA00005890
 Format
 Document (PDF)
 Title
 A computational approach analyzing global dynamics.
 Creator
 Kasti, Dinesh, Van der Vorst, R.C.A.M., Kalies, William D., Graduate College
 Abstract/Description

We describe the lattice structure of attractors in a dynamical system and the lifting of sublattices of attractors, which are computationally less accessible, to lattices of forward invariant sets and attracting neighborhoods, which are computationally accessible. We also show how the use of these algebraic structures of lattices to help us to capture the information about underlying dynamical system in a more elegant way and with lesser computational cost. For example, they can be used to...
Show moreWe describe the lattice structure of attractors in a dynamical system and the lifting of sublattices of attractors, which are computationally less accessible, to lattices of forward invariant sets and attracting neighborhoods, which are computationally accessible. We also show how the use of these algebraic structures of lattices to help us to capture the information about underlying dynamical system in a more elegant way and with lesser computational cost. For example, they can be used to develop a much efficient algorithm to compute a global lyapunov function that describes the overall gradient dynamics.
Show less  Date Issued
 2015
 PURL
 http://purl.flvc.org/fau/fd/FA00005888
 Format
 Document (PDF)
 Title
 Probabilistic and numerical validation of homology computations for nodal domains.
 Creator
 Day, Sarah, Kalies, William D., Mischaikow, Konstantin, Wanner, Thomas
 Date Issued
 20070711
 PURL
 http://purl.flvc.org/fau/fd/FAUIR000159
 Format
 Citation
 Title
 Computing Global Decompositions of Dynamical Systems.
 Creator
 Ban, Hyunju, Kalies, William D., Florida Atlantic University
 Abstract/Description

In this dissertation we present a computational approach to Conley's Decomposition Theorem, which gives a global decomposition of dynamical systems, and introduce an explicit numerical algorithm with computational complexity bounds for computing global dynamical structures of a continous map including attractorrepeller pairs and Conley's Lyapunov function. The approach is based on finite spatial discretizations and combinatorial multivalued maps. The method is successful in exhibiting...
Show moreIn this dissertation we present a computational approach to Conley's Decomposition Theorem, which gives a global decomposition of dynamical systems, and introduce an explicit numerical algorithm with computational complexity bounds for computing global dynamical structures of a continous map including attractorrepeller pairs and Conley's Lyapunov function. The approach is based on finite spatial discretizations and combinatorial multivalued maps. The method is successful in exhibiting approximations of attractorrepeller pairs, invariant sets, and Conley's Lyapunov function. We used the C++ language to code the algorithm.
Show less  Date Issued
 2006
 PURL
 http://purl.flvc.org/fau/fd/FA00000848
 Subject Headings
 Lyapunov functions, Control theory, Mathematical optimization, Differentiable dynamical systems
 Format
 Document (PDF)
 Title
 The Circular Restricted Four Body Problem is NonIntegrable: A Computer Assisted Proof.
 Creator
 Kepley, Shane, Kalies, William D., MirelesJames, Jason D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Gravitational Nbody problems are central in classical mathematical physics. Studying their long time behavior raises subtle questions about the interplay between regular and irregular motions and the boundary between integrable and chaotic dynamics. Over the last hundred years, concepts from the qualitative theory of dynamical systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles, KAM theory, and whiskered invariant tori, have come to play an increasingly important...
Show moreGravitational Nbody problems are central in classical mathematical physics. Studying their long time behavior raises subtle questions about the interplay between regular and irregular motions and the boundary between integrable and chaotic dynamics. Over the last hundred years, concepts from the qualitative theory of dynamical systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles, KAM theory, and whiskered invariant tori, have come to play an increasingly important role in the discussion. In the last fty years the study of numerical methods for computing invariant objects has matured into a thriving subdiscipline. This growth is driven at least in part by the needs of the world's space programs. Recent work on validated numerical methods has begun to unify the computational and analytical perspectives, enriching both aspects of the subject. Many of these results use computer assisted proofs, a tool which has become increasingly popular in recent years. This thesis presents a proof that the circular restricted four body problem is nonintegrable. The proof of this result is obtained as an application of more general rigorous numerical methods in nonlinear analysis.
Show less  Date Issued
 2017
 PURL
 http://purl.flvc.org/fau/fd/FA00004997
 Subject Headings
 Dissertations, Academic  Florida Atlantic University, Mathematical physics., Invariants., Dynamical systems
 Format
 Document (PDF)
 Title
 An Algorithmic Approach to The Lattice Structures of Attractors and Lyapunov functions.
 Creator
 Kasti, Dinesh, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Ban and Kalies [3] proposed an algorithmic approach to compute attractor repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a...
Show moreBan and Kalies [3] proposed an algorithmic approach to compute attractor repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a global Lyapunov function for Morse decomposition very e  cient. One of the them is to utilize highly memorye cient data structures: succinct grid data structure and pointer grid data structures. Another technique is to utilize Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is an essential step to compute a Lyapunov function. Finally, another major technique in achieving a signi cant improvement in e ciency is the utilization of the lattice structures of the attractors and attracting neighborhoods, as explained in [32]. The lattice structures have made it possible to let us incorporate only the joinirreducible attractorrepeller pairs in computing a Lyapunov function, rather than having to use all possible attractorrepeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical system allow for general algebraic treatment of global gradientlike dynamics. The separation of these algebraic structures from underlying topological structure is the basis for the development of algorithms to manipulate those structures, [32, 31]. There has been much recent work on developing and implementing general compu tational algorithms for global dynamics which are capable of computing attracting neighborhoods e ciently. We describe the lifting of sublattices of attractors, which are computationally less accessible, to lattices of forward invariant sets and attract ing neighborhoods, which are computationally accessible. We provide necessary and su cient conditions for such a lift to exist, in a general setting. We also provide the algorithms to check whether such conditions are met or not and to construct the lift when they met. We illustrate the algorithms with some examples. For this, we have checked and veri ed these algorithms by implementing on some noninvertible dynamical systems including a nonlinear Leslie model.
Show less  Date Issued
 2016
 PURL
 http://purl.flvc.org/fau/fd/FA00004668
 Subject Headings
 Differential equations  Numerical solutions., Differentiable dynamical systems., Algorithms.
 Format
 Document (PDF)
 Title
 Curve shortening in secondorder lagrangian.
 Creator
 Adams, Ronald Edward, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

A secondorder Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lowerorder derivatives play a key role in forcing certain types of dynamics. However, the application of...
Show moreA secondorder Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lowerorder derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for secondorder Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004175, http://purl.flvc.org/fau/fd/FA00004175
 Subject Headings
 Critical point theory (Mathematical analysis), Differentiable dynamical systems, Geometry,Differential, Lagrange equations, Lagrangian functions, Mathematical optimization, Surfaces of constant curvature
 Format
 Document (PDF)
 Title
 Nonlinear Phenomena from a Reinjected Horseshoe.
 Creator
 Fontaine, Marcus, Kalies, William D., Naudot, Vincent, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

A geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the standard horseshoe, but where the set of points that escape are now reinjected and contribute to richer dynamics. We show it is observed in the unfolding of a threedimensional vector field possessing an inclinationflip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic topology to...
Show moreA geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the standard horseshoe, but where the set of points that escape are now reinjected and contribute to richer dynamics. We show it is observed in the unfolding of a threedimensional vector field possessing an inclinationflip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic topology to show that for suitable parameters the flow contains a strange attractor.
Show less  Date Issued
 2016
 PURL
 http://purl.flvc.org/fau/fd/FA00004591
 Subject Headings
 Nonlinear theories., Computational dynamics., Attractors (Mathematics), Chaotic behavior in systems., Mathematical physics.
 Format
 Document (PDF)