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Tipping Points in Complex Flows -
Numerical Methods for Bifurcation Analysis of Large-Scale Systems
“The stochastic dynamical systems approach to complex flows”
In this presentation I will give an overview of what is usually called the stochastic dynamical systems approach to complex flows. In short, this approach consists of systematically investigating the stateparameter space of a particular stochastic model of a complex flow to determine the qualitative different regimes of behavior. I will outline what information of the system is needed, what computational tools are required and which collaborative efforts are needed to produce general purpose methodology. The approach will be illustrated with a model of the climate system which displays complex spatial/temporal variability.
Institute for Marine and Atmospheric Research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
e-mail: H.A.Dijkstra@uu.nl - Web page: http://www.staff.science.uu.nl/ dijks101/
“Orbit continuation for computing stable/unstable manifolds with application to the Lorenz equations and the CR3BP”
In this talk I will demonstrate the remarkable effectiveness of numerical continuation and boundary value formulations for computing stable and unstable manifolds. The first application concerns the so-called Lorenz manifold, for which the computations provide insight into the nature of the Lorenz attractor [1-3]. The second application concerns the Circular Restricted Three-Body Problem (CR3BP), which models the motion of a satellite in an Earth-Moon-like system . Specifically we compute the unstable manifold of periodic orbits known as "Halo orbits", which have been used in actual space missions. The calculations lead to the detection of heteroclinic connections from a Halo orbit to invariant tori. Subsequent continuation of such connections (as the Halo orbit is allowed to change) leads to a variety of connecting orbits that may be of interest in space-mission design. I will also mention the use of the basic algorithm for PDEs, as used in recent work by Lennaert van Veen , Bart Oldeman , and co-workers.
 E. J. Doedel, B. Krauskopf, H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity 19, 2006, 2947-2972.
 P. Aguirre, E. J. Doedel, B. Krauskopf, H. M. Osinga, Investigating
the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields, Discrete and Continuous Dynamical Systems, Vol. 29, #4, 2011, 1309?1344.
 E. J. Doedel, B. Krauskopf, H. M. Osinga, Global invariant manifolds in the transition to preturbulence in the Lorenz system, Indagationes Mathematicae, Special Issue commemorating Floris Takens, accepted, 2011.
 R. Calleja, E. J. Doedel, A. R. Humphries, B. E. Oldeman, Computing Invariant Manifolds and Connecting Orbits in the Restricted Three Body Problem, preprint, 2011.
 L. van Veen, Genta Kawahara, Matsumura Atsushi, On matrix-free computation of 2D unstable manfolds, SIAM J. Sci. Comput. 33, pp. 25-44, 2011.
 B. Borek, L. Glass, B. E. Oldeman, Continuity of Resetting a Pacemaker in an Excitable Medium, SIAM J. Applied Math., to appear.
“Transitions in magnetized plasmas: exemplary magnetohydrodynamic evolutions”
The large-scale (fluid) behaviour of astrophysical plasmas is governed by the magnetohydrodynamic (MHD) equations . I will highlight modern insights in both linear stability as well as nonlinear dynamical evolutions, where sudden transitions occur. In particular, starting from the knowledge of all wave modes (slow, Alfv´en and fast magnetoacoustic) in a homogeneous plasma, we contrast the ideal regime with a resistive counterpart, where the different wave modes are damped by Ohmic dissipation. When plasma inhomogeneity is introduced, the MHD eigenmode spectrum is organized about the frequency ranges of the continuous subspectrum, although the linear eigenvalues relocate suddenly from ideal to resistive regimes. Global unstable tearing modes can appear when reconnection is allowed by finite resistivity. I show some parametric studies of linear eigenmode computations where the role of global unstable tearing modes, in static to stationary conditions, low to high magnetization degrees, is gradually charted. I also make links with further direct numerical simulations using automated grid-adaptivity. Even planar magnetic configurations can show dramatic transitions in their overall nonlinear evolution. For current sheets unstable to tearing modes, reconnections between closely spaced antiparallel magnetic field lines may suddenly transit to more chaotic, self-feeding behavior at sufficiently large magnetic Reynolds numbers. Combined insights from linear stability analysis, to nonlinear simulations, leave much room for future plasma dynamical research.
 J.P. (Hans) Goedbloed, R. Keppens, and S. Poedts, Advanced magnetohydrodynamics.With applications to laboratory and astrophysical plasmas, Cambridge University Press, 2010.
Y. Fyodorov, F. den Hollander, S. Nechaev, H. Rootzen, S. Shlosman
R. Keppens: Centre for Plasma Astrophysics, Department of Mathematics, K.U.Leuven, Belgium
“Bistable systems with Stochastic Noise: Virtues and Limits of effective Langevin equations for the Thermohaline Circulation strength”
The understanding of the statistical properties and of the dynamics of multistable systems is gaining more and more importance in a vast variety of scientific fields. This is especially relevant for the investigation of the tipping points of complex systems. Sometimes, in order to understand the time series of given observables exhibiting bimodal distributions, simple one-dimensional Langevin models are fitted to reproduce the observed statistical properties, and used to investing-ate the projected dynamics of the observable. This is of great relevance for studying potential catastrophic changes in the properties of the underlying system or resonant behaviours like those related to stochastic resonance-like mechanisms. In this paper, we propose a framework for encasing this kind of studies and show, using simple box models of the oceanic circulation and choosing as observable the strength of the thermohaline circulation. We study the statistical properties of the transitions between the two modes of operation of the thermohaline circulation under symmetric boundary forcing and test their agreement with simplified one-dimensional phenomenological theories. We extend our analysis to include stochastic resonance-like amplification processes. We conclude that fitted one-dimensional Langevin models, when closely scrutinised, may result to be more ad-hoc than they seem, lacking robustness and/or well-posedness. They should be treated with care, more as an empiric descriptive tool than as methodology with predictive power.
Valerio Lucarini, University of Hamburg, Germany and University of Reading, UK
“How to turn an exponential into an inverse”
The construction of a bifurcation diagram requires the computation of steady states and of their stability.
The most effective ways to carry out these calculations are Newton's method and the inverse power/Arnoldi method, respectively, both of which require solving linear systems involving the Jacobian matrix.
For physical phenomena described by elliptic or parabolic partial differential equations in two or three spatial dimensions, the Jacobian matrix is both too large to be inverted directly and too poorly conditioned to be inverted iteratively.
The poor conditioning, due to the Laplacian operator, is manifested as stiffness when integrating the evolution and has led to the widespread use of implicit time-stepping methods.
This suggests using the Laplacian inversion already present in time-stepping codes (as Poisson, Stokes, or Helmholtz solvers) as an effective preconditioner.
We incorporate preconditioning by the inverse Laplacian into Newton's method and the inverse power/Arnoldi method, using BiCGSTAB to solve the resulting linear systems. Existing time-stepping codes can be easily modified to carry out either of these tasks. We describe several applications: spherical Couette flow described by the Navier-Stokes equations, Rayleigh-Benard convection described by the Boussinesq equations, and Bose-Einstein condensation described by the nonlinear Schrodinger equation.
Juan Sanchez Umbria
“Computation of Periodic Orbits and Invariant Tori in Large-Scale Dissipative Systems”
The invariant manifolds of a dynamical system are organizing centers which drive its behaviour around them. Therefore, their computation and the study of their dependence on the parameters of the system is necessary in order to understand the dynamics. The continuation of steady solutions with respect to parameters is now a common tool in Science and Engineering. The computation of other invariant manifolds as periodic orbits and invariant tori is not so usual in the case of large-scale systems, although all of them can be cast into a common framework. Their computation can be reduced to the calculation of fixed points of a map G(x,p) (x being the phase space variables and p a parameter), i.e., to the solution of an equation of the form x-G(x,p)=0.
In the case of periodic orbits, G(x,p) is the Poincare map, defined on a manifold which intersects transversally the periodic orbit. In the case of invariant tori, G(x,p) is a synthesized map which can be defined in several ways. We consider two of them one of which is trivially parallelizable If the initial system is dissipative, the differential of x-G(x,p) has all its eigenvalues clustered around +1, and therefore no preconditioner is required for the linear solvers used during the Newton's iterations.
The Newton-Krylov method to compute periodic orbits can be extended to the case of multiple shooting to try to speed up the continuation process by means of parallelism. A preconditioner for the linear systems, which appear in the application of Newton's method to the multiple shooting, must be used in this case. It is based on the information on the stability of nearby solutions, which is available from the continuation and bifurcation analysis. Therefore the preconditioner can be obtained at a low extra cost. A test problem of thermal convection of a binary fluid has been used as a test of these techniques. In the case of the multiple shooting, efficiencies close to one are attainable for low values of the number of shoots.
Juan Sanchez Umbria (in collaboration with Marta Net and Carles Simo)
“Solving large nonlinear sparse systems”
Nonlinear sparse systems invariably lead to sparse linear systems to be solved. Here, a new preconditioner for solving sparse systems arising in CFD will be presented. Until now the ingredients for the construction of incomplete LU factorizations consisted of dropping (or lumping) and ordering. Here the dropping part is a very delicate action since the preconditioner might loose nice properties which are present in the original matrix, e.g. positive definiteness. We add an extra ingredient: transformation. This opens a new world of possibilities for incomplete factorization. Using these ingredients we were able to construct a two-level block ILU that shows convergence independent of the grid size for incompressible (Navier-)Stokes equations. This factorization retains nice properties of the original matrix and robustness and convergence can be proved. This preconditioner is also very well suited to solve bordered systems. These systems occur in many places: badly posed problems, eigenvalue computations, continuation, stochastic PDEs. It will be explained how this preconditioner can be used.