Frank Schilder

Continuation of quasi-periodic invariant tori

Many systems in science and engineering can be modeled as coupled or forced nonlinear oscillators, which may possess quasi-periodic or phase-locked invariant tori. Since there exist routes to chaos involving the break-down of invariant tori, these phenomena attract considerable attention. This paper presents a new algorithm for the computation and continuation of quasi-periodic invariant tori of ODEs that is based on a natural parametrization of such tori. Since this parametrization is uniquely defined, the proposed method requires neither the computation of a base of a transversal bundle nor remeshing during continuation. It is independent of the stability type of the torus, and examples of attracting and saddle-type tori are given. The algorithm is robust in the sense that it can compute approximations to weakly resonant tori. The performance of the method is demonstrated with examples.

Recipes for Continuation

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. The book 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. It also 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. The book 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. A large number of the exercises at the end of each chapter can be used as self-study or for course assignments that range from reflections on theoretical content to implementations in code of algorithms and toolboxes that generalize the discussion in the book or the literature. Open-ended projects throughout the book provide opportunities for summative assessments. Audience: It is intended for students and teachers of nonlinear dynamics and engineering, as well as engineers and scientists engaged in modeling and simulation, and is valuable to potential developers of computational tools for analysis of nonlinear dynamical systems. It assumes some familiarity with MATLAB programming and a theoretical sophistication expected of upper-level undergraduate or first-year graduate students in applied mathematics and/or computational science and engineering. Contents: Part I: Design Fundamentals: Chapter 1: A Continuation Paradigm; Chapter 2: Encapsulation; Chapter 3: Construction; Chapter 4: Toolbox Development; Chapter 5: Task Embedding; Part II: Toolbox Templates: Chapter 6: Discretization; Chapter 7: The Collocation Continuation Problem; Chapter 8: Single-Segment Continuation Problems; Chapter 9: Multisegment Continuation Problems; Chapter 10: The Variational Collocation Problem; Part III: Atlas Algorithms: Chapter 11: Covering Manifolds; Chapter 12: Single-Dimensional Atlas Algorithms; Chapter 13: Multidimensional Manifolds; Chapter 14: Computational Domains; Part IV: Event Handling: Chapter 15: Special Points and Events; Chapter 16: Atlas Events and Toolbox Integration; Chapter 17: Event Handlers and Branch Switching; Part V: Adaptation: Chapter 18: Pointwise Adaptation and Comoving Meshes; Chapter 19: A Spectral Toolbox; Chapter 20: integrating Adaptation in Atlas Algorithms; Part VI: Epilogue: Chapter 21: Toolbox Projects; Index

An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints

This paper presents an extended formulation of the basic continuation problem for implicitly-defined, embedded manifolds in R n . The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-co-dimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand; and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user-definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation package COCO and auxiliary toolboxes, including continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.Copyright

Numerical Bifurcation of Hamiltonian Relative Periodic Orbits

Relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and the motion of rigid bodies. RPOs are solutions which are periodic orbits of the symmetry-reduced system. In this paper we analyze certain symmetry-breaking bifurcations of RPOs in Hamiltonian systems with compact symmetry group and show how they can be detected and computed numerically. These are turning points of RPOs and relative period-doubling and relative period-halving bifurcations along branches of RPOs. In a comoving frame the latter correspond to symmetry-breaking/symmetry-increasing pitchfork bifurcations or to period-doubling/period-halving bifurcations. We apply our methods to the family of rotating choreographies which bifurcate from the famous figure eight solution of the three-body problem as angular momentum is varied. We find that the family of choreographies rotating around the

Investigation of Preferred Orientations in Planar Polycrystals

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Experimental Bifurcation Analysis by Control-Based Continuation: Determining Stability

-axis bifurcates to the family of rotating choreographie...

Nonlinear effects in examples of crowd evacuation scenarios

More accurate manufacturing process models come from better understanding of texture evolution and preferred orientations. We investigate the texture evolution in the simplifled physical framework of a planar polycrystal with two slip systems used by Prantil et al. (1993, J. Mech. Phys. Solids, 41(8), 1357-1382). In the planar polycrystal, the crystal orientations behave in a manner similar to that of a system of coupled oscillators represented by the Kuramoto model. The crystal plasticity flnite element method (CPFEM) and the stochastic Taylor model (STM), a stochastic method for mean-fleld polycrystal plasticity, predict the development of a steady-state texture not shown when employing the Taylor hypothesis. From this analysis, the STM appears to be a useful homogenization method when using representative standard deviations.

Relative Lyapunov Center Bifurcations

The newly developed control-based continuation technique has made it possible to perform experimental bifurcation analysis, e.g. to track stable as well as unstable branches of frequency responses directly in experiments. The method bypasses mathematical models, and systematically explores how vibration characteristics of dynamical systems change under variation of parameters. The method employs a control scheme to modify the response stability. While this facilitates exploration of the unstable branches of a bifurcation diagram, it unfortunately makes it impossible to distinguish previously stable and unstable equilibrium states. We present the ongoing work of developing and applying the control-based continuation method to an experimental mechanical test-rig, consisting of a harmonically forced nonlinear impact oscillator controlled by electromagnetic actuators. Furthermore we propose and test ideas on how to determine the stability of equilibria states during continuation.Copyright

Exploring the performance of a nonlinear tuned mass damper

Severe accidents with many fatalities have occurred when too many pedestrians had to maneuver in too tight surroundings, as during evacuations of mass events. This demonstrates the importance of a better general understanding of pedestrians and emergent complex behavior in crowds. To this end, we develop both a new microscopic agent-based pedestrian model and also study simplified evacuation scenarios which permit the isolation of relevant nonlinear effects and their systematic investigation. We concentrate on two effects: First, the influence of the position and size of an obstacle in front of an emergency exit on the flux through the exit, and second, the influence of other pedestrians on the route choice of an individual. The first investigation demonstrates the possibility of improving substantially the flow through an exit by placing an obstacle in a suitable way in front of it. The latter shows clearly bistable states and hysteresis effects, indicating the existence of unstable pedestrian flow states in addition to the stable states. Furthermore, this set-up is an example of a radical change of the pedestrian flux by only a small change in the geometry of the evacuation scenario. The results motivate further investigation and eventually engineering use by optimizing the design of large buildings, stations, airports and stadiums for mass events.

Experimental bifurcation analysis of an impact oscillator - Tuning a non-invasive control scheme

Relative equilibria (REs) and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and rigid body motion. REs are equilibria, and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov center bifurcations are bifurcations of RPOs from REs corresponding to Lyapunov center bifurcations of the symmetry reduced dynamics. In this paper we first prove a relative Lyapunov center theorem by combining recent results on the persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov center theorem of Montaldi, Roberts, and Stewart. We then develop numerical methods for the detection of relative Lyapunov center bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian REs of the