Rigorous Shadowing of Numerical

Rigorous Shadowing of Numerical Solutions of Ordinary Di erential Equations by Containment Wayne Brian Hayes Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2001 An exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow. We present a general-purpose method for proving the existence of nite-time shadows of numerical ODE integrations of arbitrary dimension in which some measure of hyperbolicity is present and there is either 0 or 1 expanding modes, or 0 or 1 contracting modes. Much of the rigor is provided automatically by interval arithmetic and validated ODE integration software that is freely available. The method is a generalization of a previously published containment process that was applicable only to two-dimensional maps. We extend it to handle maps of arbitrary dimension with the above restrictions, and nally to ODEs. The method involves building n-cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times. The proof consists of two steps: rst, the rigorous computational veri cation of an inductive containment property; and second, a simple geometric argument showing that this property implies the existence of a shadow. The computational step is almost entirely automated and easily adaptable to any ODE problem. The method allows for the rescaling of time, which is a necessary ingredient for successfully shadowing ODEs. Finally, the method is local, in the sense that it builds the shadow inductively, requiring information only from the most recent integration step, rather than more global information typical of several other methods. The method produces shadows of comparable length and distance to all currently published results. iii Acknowledgements Our strength as a species comes from our ability to communicate with each other. Very few feats, scholarly or otherwise, can be accomplished in a vacuum. Without the ideas, help, and challenges from professional collegues, and without the warmth, care, and compassion of friends and family, our work would be non-existent or meaningless. On the professional side, I thank my committee. Their doors were always open, both for professional consultation, and for occasional personal discussion. The guidance of my supervisor, Ken Jackson, both around obstacles and out of dead ends, is much appreciated. On many occasions Wayne Enright provided much-needed guidance when my understanding of accuracy and stability issues went awry, and his keen eye for precise use of terms tightened my presentation in several places in the thesis. Tom Fairgrieve's experience with nonlinear chaotic systems helped me to view the larger picture, and his probing questions always made me think carefully about what was important. Ted Shepherd's practical experience in numerical techniques for physics problems and deep understanding of classical mechanics provided me with an even wider view than I would have ever thought possible. I cannot thank them enough. This thesis was inspired by a desire to question, and ultimately to validate, the reliability of graviational n-body integrations. Of course, that turns out to be far too large and idealistic a goal, but along the way I had the aid of several astronomers and physicists who helped keep me honest and informed about how my work related to that goal. Gerald Quinlan and Scott Tremaine, in particular, wrote the paper that ultimately led to this work, and they paid me the highest compliment that could be paid to a graduate student at the beginning of his research career: initially enthusiastic and continuing interest in my work and how it related to their own. I thank Paul Selick and Robert Corless for agreeing to be external examiners of my thesis, and for helpful comments on content and presentation of the ideas. Other professional colleagues who helped along the way include: Ned Nedialkov, for guidance into the use of his validated ODE integrator; Je Tupper, for insightful late-night discussions about interval arithmetic and philosophy; James Stewart, for convincing me that inequalities were necessary in the de nition of the inductive containment property; John Pryce, for the chance to give an impromptu talk on shadowing at a SIAM conference when a slot suddenly opened up; John Pryce and Jens von Bergmann for several days of discussions in trying to extend the proofs to arbitrary dimension; Danny House, for realizing that my problems were related to homotopy theory, and for pointing me towards Paul Selick; Jim Clarke, for putting up with an annoying young Lecturer and teaching assistant for several years; and Martha Hendriks, iv for being a den mother to all of us Comp. Sci. students. My o ce-mate, Luis Dissett, was a great companion these past years, and his incredible ability to immediately provide small proofs and counter-examples guided me through several minor emergencies. His deep yet joyfully held views on life, philosophy, mathematics, and religion provided many needed hours of distraction. I had the pleasure of lecturing several courses during my graduate career. I found the experience extremely rewarding, for I believe that there is no greater gift to a teacher than to be given the privilege to teach students who are willing to learn. Watching your faces as confusion turned to insight will be a sight I'll not soon forget. Friends and family are indirectly part of any endeavor. I would like to thank my mother, E. Merle Hayes, for single-handedly raising a ne boy (or so I'm often told), and for supporting me and encouraging me to explore the world, and for sharing that exploration, throughout my childhood, and extending into my adult life. Without her there's no telling where I'd be or what (di erent kinds of) trouble I'd be in. Each of us has a certain group of friends that are likely to last a lifetime. For me, it is the group I met as an undergraduate in the Department of Computer Science at the University of Toronto in the late 1980's and early 1990's. You are a ectionately known as the \Cabal", and you know who you are. The number of friends I've made in graduate school boggles the mind. They are too numerous to mention here. For many, the road has been long and hard, and graduation is close-at-hand, or a goal already reached. For the rest of you, the adventure has just begun! Life outside of CS must exist as well, for a balanced life. I made many friends at the University of Toronto Outing Club. Some of them even aided my research. I thank Martin and Marlene for a front seat with a map light during a long, dark drive home, so that I could work on several inspirations that occurred after a long weekend of canoeing in Killarney. Finally, I would like to thank the people of Canada, and of Ontario, for the public funding that supported my research and its publication. In the end, research is about improving people's lives and the environment in which we live. This must never be forgotten, and public funding of research must never die, for without it we will be left only with the views of corporations and governors. Of course, any remaining errors or omissions are solely my responsibility. v