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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 [4]. 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 [5], Bart Oldeman [6], and
co-workers. [1] E. J. Doedel,
B. Krauskopf, H. M. Osinga,
Global bifurcations of the Lorenz manifold, Nonlinearity 19, 2006, 2947-2972. [2] 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. [3] 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. [4] 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. [5] L. van Veen,
Genta Kawahara, Matsumura Atsushi, On matrix-free computation
of 2D unstable manfolds, SIAM J. Sci. Comput. 33, pp. 25-44, 2011. [6] 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 [1]. 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. References [1] 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. ---
“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. --- [Back] |