This book provides a comprehensive introduction to the mathematical methodology of parameter continuation, the computational analysis of families of solutions to nonlinear mathematical equations. It develops a systematic formalism for constructing abstract representations of continuation problems and for implementing these in an existing computational platform. Recipes for Continuation: Lends equal importance to theoretical rigor, algorithm development, and software engineering. Demonstrates the use of fully developed toolbox templates for single- and multisegment boundary-value problems to the analysis of periodic orbits in smooth and hybrid dynamical systems, quasi-periodic invariant tori, and homoclinic and heteroclinic connecting orbits between equilibria and/or periodic orbits. Shows the use of vectorization for optimal computational efficiency, an object-oriented paradigm for the modular construction of continuation problems, and adaptive discretization algorithms for guaranteed bounds on estimated errors. Contains extensive and fully worked examples that illustrate the application of the MATLAB?-based Computational Continuation Core (COCO) to problems from recent research literature that are relevant to dynamical system models from mechanics, electronics, biology, economics, and neuroscience.
Harry Dankowicz is Professor of Mechanical Science and Engineering at the University of Illinois, Urbana-Champaign. He is the author of a research monograph on chaos in Hamiltonian systems and a textbook on multibody mechanics, and serves as an Associate Editor of the SIAM Journal on Applied Dynamical Systems. Frank Schilder has held postdoctoral research and teaching positions at the University of Bristol, the University of Surrey, and the Technical University of Denmark. In addition to COCO, he is the author of TORCONT and RAUTO and a co-author of SYMPERCO.
Part I. Design Fundamentals: 1. A continuation paradigm; 2. Encapsulation; 3. Construction; 4. Toolbox development; 5. Task embedding; Part II. Toolbox Templates: 6. Discretization; 7. The collocation continuation problem; 8. Single-segment continuation problems; 9. Multisegment continuation problems; 10. The variational collocation problem; Part III. Atlas Algorithms: 11. Covering manifolds; 12. Single-dimensional atlas algorithms; 13. Multidimensional manifolds; 14. Computational domains; Part IV. Event Handling: 15. Special points and events; 16. Atlas events and toolbox integration; 17. Event handlers and branch switching; Part V. Adaptation: 18. Pointwise adaptation and comoving meshes; 19. A spectral toolbox; 20. Integrating adaptation in atlas algorithms; Part VI. Epilogue: 21. Toolbox projects; Index.
