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c Copyright by Nikita Agarwal May 2011 COUPLED CELL NETWORKS - INTERPLAY BETWEEN ARCHITECTURE & DYNAMICS A Dissertation Presented to the Faculty of the Department of Mathematics University of Houston In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy By Nikita Agarwal May 2011 COUPLED CELL NETWORKS - INTERPLAY BETWEEN ARCHITECTURE & DYNAMICS Nikita Agarwal Approved: Dr. Michael J. Field (Committee Chair) Department of Mathematics, University of Houston Committee Members: Dr. Matthew Nicol Department of Mathematics, University of Houston Dr. Andrew T¨or¨ok Department of Mathematics, University of Houston Dr. Jeff Moehlis Department of Mechanical Engineering, University of California, Santa Barbara, CA Dr. Mark A. Smith Dean, College of Natural Sciences and Mathematics University of Houston ii Acknowledgments First of all, I wish to thank my thesis advisor, Professor Michael Field, for his guidance, patience, generosity, and insight. I thank him for giving me freedom to explore on my own, and helping me through the difficulties in the process. I admire him for his down-to-earth attitude and simplicity. I feel fortunate to have him as my advisor. I feel immense gratitude for the academic and financial support that he provided me during my Ph.D. The research is supported by NSF Grants DMS-0600927 and DMS-0806321. I thank Dr. Matthew Nicol, Dr. Andrew T¨or¨ok, and Dr. Jeff Moehlis for being on my defense committee, reading my thesis and suggesting improvements. I thank Dr. Field for giving me the opportunity to visit University of War- wick; College of Engineering, Mathematics and Physical Sciences at University of Exeter; University of Porto; and Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) at University of Manchester, and collaborate with some great minds in the field of Dynamical Systems. These visits enhanced my knowledge and helped me in expanding my research scope through research talks and discussions. I also thank Dr. Ana Dias and Dr. Manuela Aguiar for proposing the problem discussed in Chapter 6 during my visit to Porto. I thank Dr. Peter Ashwin and Dr. David Broomhead for advising me during my visits to Exeter and Manchester, respectively. I also thank my friend and colleague Alexandre Ro- drigues (University of Porto) for insightful mathematical discussions. I hope to continue collaborating with them in future. I thank the Department of Mathematics at UH for giving me the opportu- nity to teach Honors Calculus which enhanced my mathematical knowledge and iii improved my teaching skills. My teaching supervisors, Dr. Matthew Nicol and Dr. Vern Paulsen for the encouragement. I thank the professors at UH, who have helped me in exploring various areas of Mathematics. I thank my teachers at Lady Shri Ram College, Delhi, India who laid strong mathematical background and motivated me to pursue a career in Mathematics. I thank graduate advisors, Ms. Pamela Draughn and Dr. Shanyu Ji, and other staff members for making the administrative work easier. On a personal note, I thank my family (my father Mr. Hem Prakash Agarwal, mother Mrs. Shashi Agarwal, my brothers Dr. Sidharth and Nishant, my sister- in-laws Jolly and Shweta) for their immense love and constant support, due to which I was able to write this thesis. I thank my little nephew and nieces, Manvi, Avni, Soumya, and Paras for their love; my uncle, Mr. Ramesh Agarwal for his encouragement. I am extremely fortunate to have my best friend, Ila Gupta, who has given me unconditional love and support throughout the past years. I thank her for having full faith in me even when I was hopeless, and showing me that there is light at the end of every tunnel. All these past years in the United States would have been impossible without her. I also thank my aunt, Mrs. Sarita Gupta for her encouragement and support. I thank my friend, Vandana Sharma for her cooperation. I thank all my friends and colleagues in the Mathematics department. I dedicate this thesis to my family and grandparents (Late Shri Jagdish Prasad, Late Smt Sarbati Devi, Late Shri Kashi Ram, and Smt Shakuntala Devi). iv COUPLED CELL NETWORKS - INTERPLAY BETWEEN ARCHITECTURE AND DYNAMICS An Abstract of a Dissertation Presented to the Faculty of the Department of Mathematics University of Houston In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy By Nikita Agarwal May 2011 v Abstract We are interested in studying the dynamics on coupled cell networks using the network topology or architecture. The original motivation of this work comes from the work by Aguiar et al. [4] on dynamics of coupled cell networks and Dias and Stewart [18] on equivalence of coupled cell networks with linear phase space. In this thesis, we give a necessary and sufficient condition for the dynamical equiv- alence of two coupled cell networks. The results are applicable to both continuous and discrete dynamical systems and are framed in terms of what we call input and output equivalence. We also give an algorithm that allows explicit construction of the cells in a system with a given network architecture in terms of the cells from an equivalent system with different network architecture. We provide a number of examples to illustrate the results we obtain. The dynamics of large coupled dynamical systems can be extremely difficult to analyze. One way of approaching the study of dynamics of networks is to start with a simple, well-understood but interesting small network and investigate how the simple network can be naturally embedded in a larger network in such a way that the dynamics of the small network appears in the dynamics of the larger network. This process is termed inflation, and was introduced by Aguiar et al. [4]. We give a necessary and sufficient condition for the existence of a strongly connected inflation. We also provide a simple algorithm to construct the inflation as a sequence of simple inflations. Finally, we give an example of a 3-cell coupled cell network having non-trivial syn- chrony subspaces that supports heteroclinic cycles with switching between them. vi Contents List of Figures x 1 Introduction 1 1.1 Overviewofthesis............................ 7 2 Background and Preliminaries 10 2.1 Generalities on coupled cell networks . .. 10 2.1.1 Structure of coupled cell networks: cells and connections . 11 2.1.2 Discrete and continuous coupled cell systems . .. 17 2.1.3 Passivecells........................... 21 2.1.4 Some special classes of coupled cell networks . .. 25 2.2 Generalitiesongraphs . 31 2.3 Synchronyclass ............................. 34 3 Dynamical Equivalence 37 4 Dynamical Equivalence : Networks with Asymmetric Inputs 41 4.1 Inputequivalence ............................ 41 4.2 Outputequivalence ........................... 47 vii 4.3 Examples ................................ 55 4.3.1 Systemswithtoralphasespace . 55 4.3.2 Systems with non-Abelian group as a phase space . 61 5 Dynamical Equivalence : General Networks 64 5.1 Outputequivalence ........................... 65 5.2 Inputequivalence ............................ 86 5.2.1 Splittings and connection matrices . 91 5.3 Examples ................................ 98 5.4 Universalnetwork. 100 6 Inflation of Strongly Connected Networks 106 6.1 Networks with a single input type . 117 6.1.1 Networkswithnoselfloops . 118 6.1.2 Networkswithselfloops . 126 6.2 Networks with multiple input types . 127 6.3 Examples ................................130 7 Switching in Heteroclinic Cycles 132 7.1 Introduction...............................132 7.2 Example.................................135 7.3 Switching in a heteroclinic network . 137 7.4 Forwardswitching. 143 7.5 Horseshoes................................145 8 Conclusions 147 viii Appendix 152 8.1 Relation between g and f inExample5.1.24 . 152 Bibliography 154 ix List of Figures 2.1 A cell with six inputs and one output. Inputs of the same type (for example the second and third input) can be permuted without affectingtheoutput. .......................... 11 2.2 Examples of coupled cells: (i) shows an incomplete network with an unfilled input cell A2, (ii) shows a three cell asymmetric input network;allinputsarefilled.. 13 2.3 Scalingcell ............................... 21 2.4 AddandAdd-Subtractcells . 22 2.5 Building new cells using Add, Add-Subtract cells . .... 23 2.6 Choose and pick cells C(a, b : u, v), C(1, 2:2, 2)........... 24 c1+d1 q d1 q 2.7 The lines l1 : v = u , l2 : v = u + , l3 : v = − c2+d2 − c2+d2 − d2 d2 α+β p β p u , l4 : v = u + , R1: feasibility region of (T ), R2: γ+δ − γ+δ δ δ feasibility region of (D)......................... 30 2.8 Two representations of the same directed graph . ... 32 4.1 input dominated by ........................ 42 M N 4.2 The system built using the cell from and an Add-Subtract cell. 43 G F x 4.3 The system built using 3 cells from and an Add-Subtract cell. 49 G F 4.4 Networks and differ in the first input to B ........... 55 M N 2 5.1 The cell D. The choose and pick cells are linear combinations of the vector field f modelling ...................... 79 F 5.2 The cell D, built from class C cells .................. 83 5.3 Universal network of 2 cells - (a) without self loops, (b) with self loopsallowed. ..............................103 5.4 Universal network of 3 cells. The network with edges labelled 1, , 4 ··· is a universal network without self loops; together with edges la- belled 5, 6 is a universal network with self loops allowed. 104 6.1 is a 2-fold simple inflation of ...................110 N M 6.2 Network architectures of , , . ................116 M0 ··· M9 6.3 is a strongly connected 3-fold simple inflation of ; is a N1 M N2 strongly connected 2-fold simple inflation of ............119 M 6.4 The composite of two simple inflations . 125 6.5 A strongly connected (4, 3, 2)-inflation , of (with symmetric M4 M inputs)..................................128 6.6 A strongly connected (4, 3, 2)-inflation , of (asymmetric inputs) 129 N M 7.1 The network .............................135 M 7.2 Plot shows the time evolution of x and the trajectories close to the cycles γ , i =1, , 4 projected on the (x , x )-plane.