Special Session 101: Applied Dynamical Systems in Action

Higher order approximations to the (dual) semi-geostrophic equation

Ioannis Giannoulis University of Ioannina Greece Co-Author(s):    Vasileios Kalivopoulos

Abstract: The semi-geostrophic equation is a reduced model for the approximate description of large-scale atmospheric or oceanic flows. The corresponding Lagrangian flow is measure preserving and the density of this image measure is evolved by a transport equation with a velocity field that is determined through the solution of a Monge-Ampere equation where the right hand side is the transported density. This coupled system is the so called dual semi-geostrophic equation for which the global existence of smooth solutions is still open. On the other hand, by a suitable linearization of the Monge-Ampere equation one obtains a Poisson equation and hence, in two dimensions, the dual semi-geostrophic equation is formally approximated by the Euler vorticity equation which has globally smooth solutions. G. Loeper made this approximation result rigorous to leading order on time scales of order $O(1/\varepsilon) $ with an error of order $O (\varepsilon) $. We present a generalization of this result for higher order approximations with the higher order correction terms being determined by nonlocal linear inhomogeneous transport equations. This allows us to reduce the order of the error to $O(\varepsilon^k)$ for arbitrary $k \in \mathbb{N}$. This is work supported by the Hellenic Foundation for Research and Innovation (HFRI).

Semiflow Structure in Viscous Fiber Dynamics

Thomas Hagen The University of Memphis USA Co-Author(s):

Abstract: We study a one dimensional dynamical system describing viscous fiber stretching, arising from industrial fiber spinning processes in which a highly viscous fluid is drawn into a thin filament. The model is obtained by cross sectional averaging of the axisymmetric Stokes equations with a free boundary and yields a coupled nonlinear evolution for mass transport and momentum balance dominated by viscous forces.\ Using a Lagrangian formulation, we derive a representation of the fiber cross sectional area that ensures forward invariance of positivity, rules out finite time breakup, and allows continuation of local solutions to global ones, yielding existence and uniqueness of the induced semiflow.\ The presentation addresses extensions to nonisothermal models, where temperature dependent viscosity couples the dynamics to a temperature equation and introduces a moving boundary formulation. We highlight which elements of the Lagrangian approach continue to govern global evolution and which require modification due to thermal coupling and boundary motion.

Brain Imaging and Mathematical Modeling of Alzheimer`s Disease

Peter Hinow University of Wisconsin - Milwaukee USA Co-Author(s):    Dr. Muna Aryal (North Carolina Agricultural and Technical State University), Micah Hesketh (graduate student, UWM)

Abstract: A wide variety of medical imaging methods exist for research and diagnostic procedures on the Central Nervous System (CNS), among them Magnetic Resonance Imaging (MRI) and Optical Imaging. Mathematical modeling and simulation contribute significantly to an improved understanding of image data and the intricate transport, diffusion, and metabolic processes in the brain. This is of particular importance in the study of pathologies such as Alzheimer`s Disease, which, 125 years after its description, remains poorly understood and for which only moderate success has been made in its treatment. In the first work to be presented, we investigate tracer diffusion and transport in the glymphatic system by combining optical imaging in rats with a partial differential equation of reaction-advection-diffusion type. In a second work, we collect observations on the accumulation of amyloid-$\beta$, changes in neuronal density, and a decline in cognitive performance in TgF344-AD and wild-type rats. We develop a compartmental ordinary differential equation model and determine its parameters by fitting the output to the experimental observations. In the long run, our mathematical modeling effort is intended to bridge AD research in rodent models and the human condition of AD.

Geometric properties of rough curves via dynamical systems: SBR measure and local time

Peter Imkeller Humboldt University at Berlin Germany Co-Author(s):    Olivier Pamen, Frank Proske

Abstract: We investigate geometric properties of graphs of Takagi and Weierstrass type functions, represented by series based on smooth functions. They are H\older continuous, and can be embedded into smooth dynamical systems, where their graphs emerge as pullback attractors. It turns out that occupation measures and Sinai-Bowen-Ruelle (SBR) measures on their stable manifolds are dual by ``time'' reversal. Hence absolute continuity of the SBR measure is seen to be dual to the existence of local time. The link between the rough curves considered and smooth dynamical systems can be generalized in various ways. For instance, Gaussian randomizations of Takagi curves just reproduce the trajectories of Brownian motion. Applications to regularization of singular ODE by rough signals are envisaged. This is joint work with O. Pamen (U Liverpool and AIMS Ghana) and F. Proske (U Oslo).

Analysis of a PDE model for ant trail formation

Matthias Rakotomalala Technical University of Munich Germany Co-Author(s):    Charles Bertucci, Milica Tomasevic, Oscar de Wit

Abstract: We introduce a new chemotaxis model motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic PDE system describing the evolution of the population density and the chemical signal. The key novelty lies in the transport term for the population, which depends on second-order derivatives of the chemical field. This term is derived as the limit of an anticipation-reaction mechanism for an infinitesimally small ant. We establish global existence and uniqueness of solutions, as well as the propagation of the regularity of the initial data. We then analyze the long-time behavior of the system: we prove the existence of the compact global attractor and show that the homogeneous steady state becomes nonlinearly unstable under an inviscid instability criterion. Additionally, we provide a lower bound on the dimension of the attractor. Conversely, we prove that for sufficiently small interaction strength, the homogeneous steady state is globally asymptotically stable. Finally, we present several numerical simulations illustrating the model`s dynamics.

Mathematical Analysis of the Dynamics in the Tobacco Plant v.s. Moth Interaction Cycle

Florian Rupp Kutaisi International University Germany Co-Author(s):

Abstract: Plant-herbivore systems provide a natural laboratory for nonlinear dynamical phenomena arising from feedback, adaptation, and multi-scale interaction. In this paper, we develop and analyze a mathematical model for the intra- and inter-species dynamics governing the interaction between the wild tobacco plant (Nicotiana attenuata) and the tobacco hawk moth and its caterpillars (Manduca sexta). Hereby special attention is given to plant defense mechanisms, i.e., nicotine production, attack signals to other tabaco plants, and signals to predator insects, like big-eyed bugs (several species in the family Geocoris), that feed on the caterpillars of the tobacco hawk moth. The resulting model is studied by techniques from the theory of dynamical systems such that conditions for stability and bifurcations of mixed-species equilibria are determined and the phase space dynamics are classified. Simulations round-up the picture. The model and its mathematical analysis illustrate how biochemical defense pathways translate into ecological phase portraits and provide a case study in how adaptive biological feedback adds to the insights of classical predator-prey dynamics.

Degenerate reaction diffusion systems arising in models for biofilm growth

Stefanie Sonner Radboud University Netherlands Co-Author(s):    J. Dockery, H.J. Eberl, V. Hissink Muller, J. Hughes, K. Mitra, S. Pop, R. Smeets

Abstract: Biofilms are dense aggregations of bacterial cells in moist ecosystems that are held together by a self-produced slimy matrix and are often attached to a surface. We consider mathematical models for spatially heterogeneous biofilms that are formulated as quasilinear reaction diffusion systems. Their characteristic feature is the two-fold degenerate diffusion coefficient for the biomass density comprising a polynomial degeneracy (as the porous medium equation) and a fast diffusion singularity as the biomass density approaches its maximum value. This degenerate equation is coupled to semilinear parabolic equations and/or ordinary differential equations for nutrient concentrations and additional substrates. We present results on the well-posedness and regularity of solutions for such systems on bounded and unbounded domains. For systems with immobilized nutrients the existence of traveling wave solutions has been shown. Numerical simulations are presented to illustrate the model behaviour.

Deriving a thermodynamic system from a Hamiltonian one

Johannes Zimmer TU Munich Germany Co-Author(s):    Alexander Mielke and Mark Peletier

Abstract: We reconsider the problem of coarse-graining a microscopic Hamiltonian dynamics which describes the motion of molecules to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures. Randomness enters through the initial data of a heat bath component of the original Hamiltonian system. The analysis uses some mathematical tools from the theory of dilations, which will be described at the hand of an example. The main interest is, however, to analyse the resulting system in the formulation of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). We will show that the coarse-grained version has this structure, with conserved energy, nondecreasing entropy and an Onsager operator describing the dissipation. A particular interest will be to understand how the building blocks of GENERIC arise from the microscopic Hamiltonian system.