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Summer 2024
Oberseminar Nonlinear Dynamics
Organizers
Program
Appointments only by arrangement.
| Tuesday, July 9th |
Phillipo Lappicy (Universidad Complutense de Madrid, Spain) |
An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension
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| Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation. This is a joint work with E. Beatriz.
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| Tuesday, July 16th |
Sören von der Gracht (Universität Paderborn) |
Exploring exotic symmetries to explain exotic behavior of network dynamical systems
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| Many dynamical systems models of real world processes exhibit the structure of a network consisting of nodes with connections between them. The specific interaction structure of a network can
produce remarkable dynamics beyond that of the individual nodes. Prominent examples include synchronization and highly complex branching behavior in bifurcations, phenomena that are not found
in dynamical systems without the structure of a network.
Network dynamical systems are not well understood mathematically, which makes it hard to quantify and control their behavior. The reason is that most of the established machinery of dynamical
systems theory fails to distinguish between networks and general dynamical systems. Several mathematical tools that are tailor-made for network problems have been proposed recently. Strikingly, they
have one thing in common: they exploit the algebraic nature of networks.
In this talk, I will give an overview over some recent results regarding the question which dynamical behavior and generic bifurcations are dictated by the network structure of a system. In particular,
I will illustrate how structural and algebraic properties culminate in symmetries of the governing equations and how these can be exploited for (partial) answers. This includes
classical symmetries but also more exotic concepts such as monoid and quiver representations.
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Hans Engler (Georgetown University) |
The Lorenz System of 1996
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| The meteorologist and applied mathematician Edward Lorenz is famous for discovering chaotic behavior in dynamical systems in 1963. In 1996, Lorenz introduced a dynamical system that describes very simple "weather" on a cartoon planet: a scalar quantity evolves on a circular array of N sites, undergoing radiative forcing, dissipation, and nonlinear advection. Lorenz proposed this system as a test bed for numerical weather prediction. Since then, it has found much use as a test case in data assimilation. Related systems have been studied by other authors earlier.
Mathematically, this is an nonlinear N-dimensional dynamical system that is invariant under rotating the sites. There is a single parameter, namely forcing strength. For small forcing strength, there is no "weather": the only possible stable solutions are constant in space and time. As the strength of the forcing increases, periodic wave patterns appear that move around the circle of sites. These periodic patterns are not unique - the same forcing strength may be associated with stable patterns that are qualitatively different, depending on the system's initial state. For even larger forcing, the motion becomes chaotic. Regular wave patterns are replaced by moving irregular wave trains that are short-lived, similar to changing weather systems that move across a landscape.
The talk will introduce the main properties of this and of related systems. The appearance of periodic solutions can be explained with bifurcation theory for all these versions. By using the discrete Fourier transform and explicit computation of normal form coefficients, the stability of bifurcating periodic solutions and the coexistence of multiple such solutions for the same radiative forcing can be understood. The system also shows delayed transition to instability as the forcing parameter increases slowly.
This is joint work with John Kerin, a former Georgetown University undergraduate student.
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Time and Place
Talks usually take place on Tuesday at 3:15 p.m.
at the Free University Berlin
Arnimallee 7 (rear building), room 140.
WIAS, Mohrenstr. 39, Erhard-Schmidt lecture room, 10117 Berlin.
The event is also accessible via Zoom.
Guests are always welcome !
Archive
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Last change: Aug. 1, 2024
