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Pitchfork Bifurcations of Invariant Manifolds

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Pitchfork Bifurcations of Invariant Manifolds 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. ASAC ICOK IUCAIOS O IAIA MAIOS b t Chpnrr I a aamee eee yamica sysem we e quaiaie sucue o e souios cages ue o a sma cage i e aamee e sysem is sai o ae uegoe a iucaio iucaios ae ee cassiie o e asis o e ooogica oeies o ie ois a iaia maios o yamica sysems A icok iucaio i is sai o ae occue we a sae ie oi ecomes usae a wo ew sae ie ois seaae y e usae ie oi come io eisece I is esis a icok iucaio o a (m-1-imesioa iaia sumai -o o a yamica sysem i m is eie aaogous o a i Suicie coiios o suc a iucaio o occu ae sae a eisece o e iucae maios is sow ue e sae yoeses e yamica sysem is assume o e a cass C1 ieomoism o eco ie i m e eisece o ocay aacig iaia maios M+ a M_ ae e iucaio as ake ace is oe y cosucig a ieomoism o e usae maio M eciques use o oig e aoe meioe esu ioe ieeia ooogy a aaysis a ae aae om ama [1] a isc [19] e mai eoem o e esis is iusae y meas o a caoica eame a aie o a -imesioa iscee esio o e oka-oea moe esciig yamics o a eao-ey ouaio e oka-oea moe is sigy moiie o ee o a coiuousy ayig aamee Sigiicace o a icok iucaio i e oka-oea moe is iscusse wi esec o ouaio yamics asy imicaios o e eoem ae iscusse om a maemaica oi o iew ICOK IUCAIOS O IAIA MAIOS b t Chpnrr A rttn Sbttd t th lt f r Inttt f hnl nd tr, h Stt Unvrt f r — r n rtl lfllnt f th rnt fr th r f tr f hlph n Mthtl Sn prtnt f Mthtl Sn, I prtnt f Mtht nd Cptr Sn, trr At 2004 Coyig © y yoi Camaeka A IGS ESEE AOA AGE ICOK IUCAIOS O IAIA MAIOS t Chpnrr eis ackmoe isseaio Aiso ae oesso eame o Maemaica Scieces I oe Miua Commiee Meme ae oesso eame o Maemaica Scieces I wee Mose Commiee Meme ae oesso eame o Maemaics a Comue Sciece uges emeiusageogiou Commiee Meme ae oesso eame o Maemaica Scieces I Amiaa ose Commiee Meme ae Associae oesso eame o Maemaica Scieces I IOGAICA SKEC Athr: yoi Camaeka r: oco o iosoy Undrrdt nd Grdt Edtn: • oco o iosoy i Maemaica Scieces ew esey Isiue o ecoogy ewak • Mase o Sciece i Aie Maemaics ew esey Isiue o ecoogy ewak • Mase o Sciece i Maemaics Uiesiy o omay omay Iia 199 Mjr: Maemaica Scieces lvn hbnd Abhjt fr nprn v ACKOWEGMES I am indebted to my advisor Professor Denis Blackmore for his untiring guidance and patience. I thank him for his support and encouragement and for being a wonderful teacher. I greatly appreciate his generosity with time and his kindness. Special thanks are due to Professor Daijit Ahiuwalia for his guidance, support and advice all through my stay at NJIT and for having confidence in me when things were bleak. I am grateful to Professor Robert Miura for agreeing to be on my committee, for carefully reading the thesis and for his corrections and suggestions. I am extremly grateful to him for his encouragement and support during my most trying times. I am very thankful to Professor Lee Mosher for being on my committee, reading the technical parts of my thesis thoroughly and rendering his valuable mathematical advice and also for his J. suggestions. Sincere thanks to Professor Demetrius Papageorgiou for being on my committee, for his advisement throughout the graduate program and for his encouragement during my graduating semester. I would also like to thank Professor Amitabha Bose for agreeing to be on my committee and for teaching me during my initial semesters at NJIT. I take this opportunity to express my deep gratitude to Professor S. Kumaresan, University of Bombay for teaching me during my masters and showing me the beauty of Mathematics. I thank him for being a marvelous teacher, for setting an example of himself and for inspiring me to pursue Mathematics. I thank Professor G. Kriegsmann, Professor D. Papageorgiou, Professor R. Miura, Professor M. Siegel, Professor J. Luke, Professor N. Farzan, Professor John Loftin (Rutgers), Professor Sharad Sane (University of Bombay), Professor Nirmala v Limaye (University of Bombay) and Professor Anjana Prasad (University of Bombay) who have taught me and encouraged me during my student years. My gratitude to the office staff - Ms. Padma Gulati, Ms. Susan Sutton, Ms. Liliana Boland and Ms. Sherri Brown for their enthusiastic and warm support and for making lives easier for us graduate students. I also take this opportunity to thank everyone at the Graduate Students Office, the Office for International Students & Faculty for helping with all the administrative complexities. I am much grateful to the Library staff at NJIT and at Columbia University for their help and for working on weekends. Thanks to my seniors Eliana Antoniou, Lyudmyla Barannyk, Urmi Gosh-Dastidar, Adrienne James, Stephen Kunec, Said Kas-Danouche, Knograt Savettaseranee, Tetyana Segin and Hoa Tran for sharing their experiences and their suggestions. I am very grateful to my friends Lin Zhou, Muhammad I. Hameed, Arnaud Goullet and Yuriy Mileyko for their lengthy discussions, support and for sharing their knowledge. I thank them and Christina Ambrosio, Oleksander Barannyk, Madhushree Biswas, Soumi Lahiri, and Tsezar Seman, for giving me the pleasure of their company. I thank my father Mr. Harbhajan Anand and my mother Dr. Shashi Anand for being my first teachers, for nurturing me and showering me with their love and encouragement. Thanks also to my affectionate brothers Ravi and Harsh for their love and support. Many thanks to my in-laws, the entire Champanerkar family for their constant support and understanding. Special thanks to Anagha for her trust, love and affection for so many years. Lastly, I thank my beloved husband Abhijit Champanerkar for his support, encouragement and infinite patience all through my graduate school. I also thank him deeply for teaching me perseverance when I most needed it. vii AE O COES Chptr Page 1 INTRODUCTION 1 1.1 Bifurcations 1 1.2 Pitchfork Bifurcation 4 1.3 About the Lotka-Volterra Model 9 1.4 Organization of the Thesis 10 2 MATHEMATICAL PRELIMINARIES 11 2.1 Basic Definitions 11 2.2 Theorems 13 2.3 New Definitions 14 3 PITCHFORK BIFURCATIONS - DISCRETE CASE 17 3.1 Notation 17 3.2 Pitchfork Bifurcations of Invariant Manifolds 19 4 ILLUSTRATION AND APPLICATION OF THE PITCHFORK BIFURCATION THEOREM 37 4.1 A Canonical Example 37 4.2 An application - Discrete Lotka-Volterra Model 42 5 PITCHFORK BIFURCATIONS - CONTINUOUS CASE 45 5.1 Pitchfork Bifurcations of Invariant Manifolds 45 6 CONCLUSIONS 53 REFERENCES 57 v IS O IGUES r Page 1.1 Bifurcation diagram for a pitchfork bifurcation in 4 1.2 Dynamics near a fixed point in RA before a bifurcation 6 1.3 Dynamics near fixed points in IR A after a pitchfork bifurcation. 6 2.1 Pitchfork bifurcation in R. 13 2.2 Locally attracting SA in Ice before a pitchfork bifurcation 15 2.3 Three diffeomorphic copies of S A in Il83 after a pitchfork bifurcation . 15 2.4 Bifurcation through non-manifold structures. 16 3.1 Illustration of Ft, = (fm , gm) 18 3.2 Regions Kip) and A. 20 4.1 Plot of r vs for the canonical example. 39 4.2 Plot of p. vs for the canonical example 40 4.3 Plot of r vs i for the canonical example (with grid) 40 4.4 Si is locally attracting 41 4.5 Any trajectory outside converges to Si 41 4.6 Any trajectory inside converges to S i 41 4.7 Si is locally repelling 41 41 4.8 Any trajectory outside converges to Si+- 4.9 Any trajectory inside converges to S i _, 41 6.1 Pitchfork bifurcation in as an intersection of two smooth curves . 54 6.2 Pitchfork bifurcation in R A as an intersection of two smooth surfaces . 55 x CAE IOUCIO . frtn A yamica sysem ca e oug o as a ow o ois i a sace o a ecusie ucio om a sace o ise A yamica sysem cages wi ime a e sae o e sysem a ay gie ime ees o is eious sae Mos maemaica moes esciig e sae o a aua sysem ee o seea aamees A yamica sysem eeig o a aamee is sai o uego a iucaio we e ooogica aue o e souio o e sysem cages quaiaiey ue o a cage i e aamee a is e ase oais o e yamica sysem eoe a ae e iucaio ae o ooogicay equiae e aue o e aamee a wic suc a quaiaie cage occus is kow as a iucaio aue e iea o acig o souios was kow o Ca acoi (13 u e em iucaio was is use y ei oicae (15 i is essay "Su equiie ue masse uie aimée u moueme e oaio" [3] oicaé eoe e eaiosi ewee saiiy a iucaio Geoge iko eee e esus o iucaios i 197 [] e use eciques om ooogy o suy yamica sysems I e 193s Aeksa Aoo a is coaoaos woke o iucaios i ei ogam - oiea osciaios eoy Eea o a Ku Oo ieics woke i iucaio eoy i e 19s I e 19s Soomo esce asae Aoos wok mae
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