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Habilitation HABILITATION The AnT project: On the simulation and analysis of dynamical systems Presented to the faculty of Computer Science, Electrical Engineering, and Information Technology at the University of Stuttgart by Dr. rer. nat. Michael Schanz Department Image University of Understanding Stuttgart Institute of Parallel and Distributed Systems In Memoriam Erika Sonja Schanz 24. M¨arz 1928 - 24. April 1993 Paul Karl Schanz 13. Januar 1926 - 24. Mai 2001 Preface A software project like the one presented in this work could never have been done without the help and support of many, many people. Among them are people who are more or less directly involved in the project as well as others which supported me in many different ways. The people I want to thank here are in fact so many, that it is not possible to thank them all personally. Therefore, I would like to thank in general the members of the Institute of Parallel and Distributed Systems (IPVS) at the University of Stuttgart, where the AnT project resides and where most of the work was done. However, some persons I want to thank personally: • The referees of this work, Prof. Bestehorn (Cottbus), Prof. Bungartz (Stuttgart), Prof. Friedrich (M¨unster)and Prof. Levi (Stuttgart). Moreover, I would like to thank Prof. Levi, head of the department Image Understanding at the IPVS, for his friendly support, for the many discussions and for giving me the opportunity and time to develop and maintain the AnT project and to write this habilitation. • All the people involved in the AnT project, but especially my friends: - Viktor Avrutin, for his brilliant ideas, profound knowledge and never ending interest in nonlinear dynamics, the uncountable valu- able discussions with him and finally his affectation for ORIGAMI. - Robert Lammert, for his creativity, his art of software engineering and his effort not only during the development of the graphical user interface but in many other parts and modules of the software. - Georg Wackenhut, for his support, his holistic thinking, his en- durance with the build mechanism especially in critical phases of the project and finally his careful preparations of the distributions. 1 Additionally I would like to thank: • Our secretary Ute Gr¨ater for her friendship and her tremendous help in nearly every case and under nearly any circumstances. • The complete RoboCup team at our institute, but especially Georg Kindermann, Reinhard Lafrenz and Frank Schreiber for the many in- teresting discussions during our many famous breakfasts. • My long standing friend Martin Ossig who supervised together with me some diploma theses. • My friends at basketball who helped me ease the built-up tension in my shoulders and neck caused by the many hours sitting in front of the computer. Here I would like to add a special thanks to Uwe Husmann, who worked with the software and found some bugs and Elmar Groß who is directly involved in the new web front end of the project. • Last but not least, my friends and former colleagues from the physics department and especially the organizer Robert H¨onlinger for the long lasting and always interesting events called ”Kaffeerunde”. Furthermore, a special thanks to my sister Martina Frey and my brothers Joachim and J¨urgenSchanz and their families for their understanding, their personal support and the many funny parties and other family events. Finally I want to thank my wife Cordula Ernst for her irresistible love, her amazing spirit, her great humor, her endless care and support and her courage for marrying me. Stuttgart, July 2004 Michael Schanz 2 Glossary s: state vector . .44 Γs: state space . 44 Ns: state space dimension . .44 si: i-th state component or state variable . 44 p: parameter vector . 44 Γp: parameter space . 44 Np: parameter space dimension . 44 pi: i-th parameter ....................................................44 s(tn): state vector at time tn ............................................48 tn: discrete time at timestamp n .....................................48 f: vector field .......................................................48 sn: state vector at timestamp n ......................................49 n: timestamp n ......................................................49 ∗ xi : fixed points . .50 αi: i-th flip bifurcation point . 51 α∞: accumulation point . 51 λ: Lyapunov exponent . .53 3 S: state vector of a CML or CODEL . .60 F : vector field of a CML or CODEL . 60 i sn: state of cell i of a CML ...........................................60 s(t): state vector at continuous time t ..................................68 t: continuous time . 68 S: state vector of a CML or CODEL . .73 F : vector field of a CML or CODEL . 73 si(t): state of cell i of a CODEL . 73 τ: delay time ........................................................82 τi: delay times .......................................................84 F: functional . 88 st: vector function on the delay interval . 88 ∆t: step size . 90 t0: initial time . 90 s(t0): initial value . .90 Rn: n dimensional Euclidean space . 90 qk: k-th scan point . 98 qin: initial scan point . 98 qfi: final scan point . 98 ∆q: scan point increment . 98 N s: number of scan points . 98 qk: k-th 2D-scan point . 100 qin: initial 2D-scan point . .100 4 qfi: final 2D-scan point . 100 ∆q: 2D-scan point increment . .100 k qi : i-th component of the k-th 2D-scan point . 100 off qi : offset of the i-th component of the 2D-scan point . 100 ai: parameter of the scan ellipse . .100 trig1: cos ..............................................................100 trig2: sin ..............................................................100 ϕk: angle of the k-th 2D-scan point . 100 ∆ϕ: 2D-scan point angle increment . 100 ϕin: angle of the initial 2D-scan point . 100 ϕfi: angle of the final 2D-scan point . .100 in qi : i-th component of the initial 2D-scan point . 100 fi qi : i-th component of the final 2D-scan point . .100 A: attractor . 111 A: wave number vector . 111 (s, m, e): floating point number representation . .224 b: bias .............................................................224 e: exponent . 224 ei: exponent bit . .224 m: mantissa . 224 mi: mantissa bit . 224 s: sign bit . .224 M: matrix ..........................................................227 5 p(λ): characteristic polynomial . 227 a λi : analytically calculated i-th eigenvalue . .227 n λi : numerically calculated i-th eigenvalue . .227 vi: eigenvector i .....................................................228 m: particle mass . 280 f : friction constant . 280 k : spring constant . 280 6 Summary In this work, a the software project AnT 4.669 is presented. This project is about a simulation and analysis tool for dynamical systems, whereby it was aimed right from the start, that not only a broad spectrum of dynamical systems is supported, but many investigation methods as well. This thesis is structured as follows: In the Introduction (Chap. 1) a motivation and an overview about the main topics and the involved scientific disciplines is given and among others, the notions of simulation and analysis and their prerequisites are defined and described. The modeling and especially the mathematical modeling are con- sidered in more detail as well as the tasks of interactive and non-interactive simulation. The AnT project chapter provides an overview about the software - its aims, requirements and features. Some historical and techni- cal remarks are given together with some basic principles and facts about the functionality and a short guidance how to use this software and its fea- tures. In the third chapter, denoted as Supported classes of dynamical systems, the notion of a dynamical system is defined and described in this context and the most important classes of dynamical systems supported by the AnT 4.669 software package including their main characteristic prop- erties are listed. For each addresses class of dynamical systems at least one typical and illustrative representative is presented. In the fourth chapter - Scanning of dynamical systems - the notion of scanning is described and its importance in the field of nonlinear dynamics is emphasized. The differ- ent types of scan possibilities and procedures provided by the software.
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