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High-Order Adaptive Methods for Computing Invariant Manifolds of Maps

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High-Order Adaptive Methods for Computing Invariant Manifolds of Maps New Jersey Institute of Technology Digital Commons @ NJIT Dissertations Electronic Theses and Dissertations Spring 5-31-2011 High-order adaptive methods for computing invariant manifolds of maps Jacek K. Wrobel New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/dissertations Part of the Mathematics Commons Recommended Citation Wrobel, Jacek K., "High-order adaptive methods for computing invariant manifolds of maps" (2011). Dissertations. 267. https://digitalcommons.njit.edu/dissertations/267 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Dissertations by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ABSTRACT HIGH-ORDER ADAPTIVE METHODS FOR COMPUTING INVARIANT MANIFOLDS OF MAPS by Jacek K. Wr´obel The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to de- crease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps. HIGH-ORDER ADAPTIVE METHODS FOR COMPUTING INVARIANT MANIFOLDS OF MAPS by Jacek K. Wr´obel A Dissertation Submitted to the Faculty of New Jersey Institute of Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematical Sciences Department of Mathematical Sciences, NJIT May 2011 Copyright c 2011 by Jacek K. Wr´obel ALL RIGHTS RESERVED APPROVAL PAGE HIGH-ORDER ADAPTIVE METHODS FOR COMPUTING INVARIANT MANIFOLDS OF MAPS Jacek K. Wr´obel Roy H. Goodman, Dissertation Advisor Date Associate Professor, Department of Mathematical Sciences, NJIT Denis L. Blackmore, Committee Member Date Professor, Department of Mathematical Sciences, NJIT Amitabha K. Bose, Committee Member Date Professor, Department of Mathematical Sciences, NJIT Richard O. Moore, Committee Member Date Associate Professor, Department of Mathematical Sciences, NJIT Lee Mosher, Committee Member Date Professor, Department of Mathematics & Computer Science, Rutgers University IOGAICA SKEC Athr: K. Wróbl r: tr f hlph t: M 20 Undrrdt nd Grdt Edtn: • tr f hlph n Mthtl Sn, r Inttt f hnl, r, , 20 • Mtr f Sn n Mtht, AG Unvrt f Sn nd hnl, Kr, lnd, 200 Mjr: Mthtl Sn rnttn nd bltn: K. Wróbl nd . Gdn, "hrrdr Mthd fr Cptn Invrnt Mnfld f Mp," n prprtn. . Gdn nd K. Wróbl, "hrrdr tn Mthd fr Cptn Invrnt Mnfld f 2 Mp," International Journal of Bifurcation and Chaos, ptd, 200. K. Wróbl nd . Gdn, "Adptv Mthd fr Cptn Invrnt Mnfld f Mp," SIAM Annl Mtn (A0, l 26, 200, ttbrh, A. vld ldrv nd Wróbl, "Mdl f trbll nr, nttn rtt n d," Studies in Computational Intelligence 180, Mdl. n. n r & S., Sprnr, rln, dlbr, 200, W. Mt, . Kprz (d., 22. K. Wróbl nd . Gdn, "Adptv Mthd fr Cptn Invrnt Mnfld f 2 Mp," SIAM Cnfrn n Appltn f nl St (0, M 2, 200, Snbrd, U. K. Wróbl nd . Gdn, "rvl ndrn rtr Crv," SIAM Cnfrn n Cpttnl Sn nd Ennrn (CSE0, Mrh 26, 200, M, . v Jacek K. Wr´obel and Roy H. Goodman, “Adaptive Method for Computing Invariant Manifolds of 2D Maps,” Applied Math Days, Rensselaer Polytechnic Institute, Oct 30-Nov 1, 2008, Troy, NY. v To My Parents vi ACKNOWLEDGMENT First I would like to express my deepest appreciation to my dissertation advisor, Professor Roy H. Goodman, for his valuable guidance, support, encouragement, kindness and en- thusiasm during the course of this work and my entire graduate study at NJIT. I am especially grateful for his inexhaustible patience. I have been greatly influenced by his organized and disciplined style of work. He has made a significant difference in my life and I will never forget it. I am very grateful to Prof. Goodman and his wife, Sarah Berger, for their hospitality during my visit at the Technion-Israel Institute of Technology. I wish to thank the committee members, Profs. Denis L. Blackmore, Amitabha K. Bose, and Richard O. Moore of the Department of Mathematical Sciences, NJIT, and Prof. Lee Mosher of the Department of Mathematics & Computer Science, Rutgers University. For helpful discussions I would like to thank Denis Blackmore, Josh Carter, Gershon Elber, Martin Hering-Bertram, Rafael de la Llave, Hector Lomel´ı, Jason Mireles-James, James Meiss, Hinke Osinga, and Denis Zorin. NJIT undergraduates Casayndra Basarab, Fatima Elgammal, Matthew Peragine and Priyanka Shah collaborated on a preliminary version of this work as part of the CSUMS program, supported by NSF DMS-0639270. The author received support from NSF DMS-0807284. vii TABLE OF CONTENTS Chapter Page 1 INTRODUCTION ................................. 1 1.1 Iterated Maps ................................. 1 1.2 Background .................................. 3 2 HIGH-ORDER ADAPTIVE METHOD FOR COMPUTING ONE-DIMENSIONAL INVARIANT MANIFOLDS OF MAPS ......... 8 2.1 Model Curves ................................. 8 2.1.1 Parametric curves ........................... 8 2.1.2 Parameterization ............................ 10 2.2 2D Geometric Modeling Tools ........................ 10 2.2.1 Piecewise Linear Interpolation .................... 10 2.2.2 B´ezier Curves ............................. 11 2.2.3 Hermite Interpolating Polynomials .................. 16 2.2.4 Catmull-Rom Splines ......................... 17 2.2.5 B-splines ................................ 20 2.2.6 Quasi-Interpolation Schemes ..................... 23 2.3 Adaptive Methods ............................... 25 2.3.1 Existing Methods ........................... 25 2.3.2 Proposed Methods ........................... 31 2.4 Numerical Tests of the Proposed Tools ................... 35 2.4.1 A Model Curve ............................ 35 2.4.2 A Visual Test and Motivation for Improved Methods ........ 36 2.4.3 Quantitative Comparison of Methods ................ 38 2.5 Details of the Numerical Implementation: One-dimensional Invariant Manifold Calculation .................................. 40 2.5.1 Inductive Parameterization ...................... 40 2.5.2 Notation ................................ 41 viii TABLE OF CONTENTS (Continued) Chapter Page 2.5.3 The Initial Primary Segment ..................... 42 2.5.4 Resolving a Simple Primary Segment ................ 44 2.5.5 Kink Patching ............................. 44 2.6 Numerical Tests ................................ 46 2.6.1 Example 1: H´enon Map ........................ 46 2.6.2 Example 2: McMillan Map ...................... 48 2.6.3 More Direct Convergence Tests .................... 51 2.6.4 Example 3: Map with an Explicit Manifold ............. 53 2.7 Discussion ................................... 55 3 HIGH-ORDER ADAPTIVE METHOD FOR COMPUTING TWO-DIMENSIONAL INVARIANT MANIFOLDS OF MAPS ......... 58 3.1 Existing Methods ............................... 58 3.1.1 Parameterization Method ....................... 59 3.1.2 The Krauskopf-Osinga Method .................... 60 3.2 3D Geometric Modeling Tools ........................ 63 3.2.1 Parametric Surface .......................... 63 3.2.2 Triangulation ............................. 64 3.2.3 Barycentric Coordinates ........................ 67 3.2.4 Bernstein Polynomials ......................... 68 3.2.5 Triangular Bernstein-B´ezier Patches ................. 69 3.2.6 Surface Normal Estimation ...................... 74 3.3 Triangulation Based Surface Approximation Schemes ........... 75 3.3.1 Bivariate Linear Interpolation .................... 75 3.3.2 Nine-parameter Interpolant ...................... 76 3.3.3 The Clough-Tocher Method ...................... 77 3.3.4 The Shirman-Sequin Method ..................... 82 ix TABLE OF CONTENTS (Continued) Chapter Page 3.3.5
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