Minisymposium NDNS+

Multiscale Dynamical Systems and Solutions

Organizer & Chair: Michiel Renger (Technische Universiteit Eindhoven)
Waaier 1, 14:15 - 16:15

Discrete-time dynamics on a space of measures

Sebastiaan Janssens (Universiteit Utrecht)

Consider a population of agents (e.g. cells or biological oscillators such as integrate-and-fire neurons) whose individual state (i-state) corresponds to a point moving on the one-dimensional unit circle S. Assume furthermore that interaction between agents is indirect, via contribution to and dependence on a so-called environmental condition. Then the population state (p-state) at a particular moment is given by a measure μ ∈ M + (S), the cone of finite positive Borel measures on S. Depending on the presence of birth and death effects, μ may or may not be a probability measure.

We investigate the existence, stability and bifurcation of Dirac-type periodic solutions of the p-state dynamics by studying the fixed points of an associated return map defined on M + (S). The underlying interpretation is that a stable (unstable) Dirac fixed point corresponds to a synchronised (de-synchronised) p-state.

Gradient Flow Model for Osmotic Cell Swelling

Martijn Zaal (Vrije Universiteit Amsterdam)

A basic model for cell swelling by osmosis is constructed, resulting in a free boundary problem. For radially symmetric initial conditions, this model can be formulated as a gradient flow on a metric by choosing a suitable pair of functional and metric. This particular choice does not require the osmotic force to be included in the formulation explicitly. It appears that this result can be generalized to non-symmetric initial conditions.

Gradient theory for plasticity as the Γ-limit of a nonlinear dislocation energy

Lucia Scardia (Technische Universiteit Eindhoven)

Since the motion of dislocations is regarded as the main cause of plastic deformation, a large literature is focused on the problem of deriving plasticity theories from more fundamental dislocation models. Although a dislocation is a lattice defect, in most dislocation models it has been described in the framework of a continuum theory, in which the positions of the atoms are averaged out. Indeed this reduces enormously the total number of degrees of freedom: From all atom positions to a few geometric quantities (displacement/deformation, dislocation line, slip planes, etc.).

The starting point of our derivation is also a continuum dislocation model. The main novelty of our approach is that we consider a nonlinear dislocation energy, whereas most mathematical and engineering papers treat only a quadratic dislocation energy, so that the constitutive relation between stress and strain is linear. Clearly, the linear constitutive relation is not satisfactory close to the dislocations’ cores, where the strains are too large for the linear approximation to hold. Moreover, the quadratic dislocation energy blows up at a dislocation, and an ad hoc parameter needs to be introduced, the so called core radius, representing the size of the region around the dislocation which needs to be removed in order to have a finite strain energy.

Our choice of a nonlinear dislocation energy allows us to define the strain energy in the whole domain, hence also close to the dislocations. Moreover, the Γ-limit of the nonlinear dislocation energy as the length of the Burgers vector tends to zero has the same form as the Γ-limit obtained by starting from a linear, semi-discrete dislocation energy, but now obtained without resorting to the introduction of an ad hoc cut-off radius. The nonlinearity, however, creates severe mathematical difficulties, which we tackled by proving suitable versions of the Rigidity Estimate in non-simply-connected domains and by performing a rigorous two-scale linearisation of the energy around an equilibrium configuration.

Desertification: instabilities of patterns

Sjors van der Stelt (Universiteit van Amsterdam / CWI)

Semi-arid ecosystems, ecosystems with an annual precipitation of about 250-500 mm, are typically found at the borders of deserts around the world. They cover about 30% of the emerged surface of the earth. Aerial photographs of many of these ecosystems show interesting regular and irregular patterns that vary in scale and shape and remind the observer of similar patterns known from the skin of panthers or zebras.

In this talk, we introduce a reaction diffusion model through which we gain insight in the emergence and disappearance of these patterns. In particular, we will discuss the possible instabilities through which spatially periodic patterns disappear and through which the ecosystem will suddenly change into a desert.

Bifurcations and stability analysis of planar Hopf-transversal systems

Xia Liu (Rijksuniversiteit Groningen)

Discontinuous vector fields, or Filippov vector fields, find applications in several fields including mechanical engineering, electrical engineering, and ecology. In this work we consider planar Filippov vector fields which consist of two smooth vector fields separated by a discontinuity boundary. One of the vector fields goes through a Hopf bifurcation while the other one is constant, transversally crosses the boundary. This Filippov system, called Hopf-transversal system, depends on three parameters. We study the Hopf-transversal system and determine its bifurcations and its phase portraits. Furthermore, we prove persistence of these bifurcations.

A scalable Helmholtz solver combining the deflation with shifted Laplace preconditioner

Abdul H. Sheikh (Technische Universiteit Delft)

The Helmholtz equation appears in the diverse phenomena such as elastic waves in solids, sound and acoustic waves, electromagnetic waves and seismic waves. Our object is to develop high performance iterative solution algorithm for solving the discrete indefinite Helmholtz equation modeling wave propagation on large scale for e.g. seismic waves for mining.

- Δu - k2u = g on boundary Ω

Ingredients in our work are the shifted Laplace preconditioner and deflation. The development of the shifted Laplace preconditioner for the Helmholtz equation was a breakthrough in the development of efficient solution techniques for the Helmholtz equation. The distinct feature of this preconditioner is the introduction of a complex shift, effective introducing damping of wave propagation in the approximate solve. This preconditioner was extensively discussed in various texts and applied in a number of different contexts. Although performant, the resulting algorithm is not truly scalable. The bigger the wavenumber, the more spectrum scatters away from one, hampering the convergence. Idea of projection has been used since long to deflate unfavorable eigenvalues. By inducting eigenvectors corresponding to unwanted eigenvalues, better convergence for CG and GMRES has been reported in various texts. We also combine the idea of deflation with shifted Laplace preconditioner, which leads to a scalable Helmholtz solver, in the sense iterations does not depend upon parameters. We provide a convergence analysis. We perform a Fourier two-grid analysis of one-dimensional model problem with Dirichlet boundary conditions discretized by a second order accurate finite difference scheme. The components analyzed are the shifted Laplace preconditioner used as smoother, full-weighting and linear interpolation inter-grid transfer operators, and a Galerkin coarsening scheme. This Fourier analysis results in a closed form expression for the eigenvalues of the two-grid operator. This expressions shows that the spectrum is favourable for convergence of Krylov subspace methods. We apply the deflated shifted Laplace preconditioner to two-dimensional model problems method with constant and non-constant wave numbers and Sommerfeld boundary conditions discretized by second order accurate finite difference scheme on uniform meshes. Numerical results show that the number of GMRES iterations is wave-number independent.