Periodic solutions to a mean-field model for electrocortical activity

We consider a continuum model of electrical signals in the human cortex, which takes the form of a system of semilinear, hyperbolic partial differential equations for the inhibitory and excitatory membrane potentials and the synaptic inputs. The coupling of these components is represented by sigmoidal and quadratic nonlinearities. We consider these equations on a square domain with periodic boundary conditions, in the vicinity of the primary transition from a stable equilibrium to time-periodic motion through an equivariant Hopf bifurcation. We compute part of a family of standing wave solutions, emanating from this point.Comment: 9 pages, 5 figure

Transitions in large eddy simulation of box turbulence

One promising decomposition of turbulent dynamics is that into building blocks such as equilibrium and periodic solutions and orbits connecting these. While the numerical approximation of such building blocks is feasible for flows in small domains and at low Reynolds numbers, computations in developed turbulence are currently out of reach because of the large number of degrees of freedom necessary to represent Navier-Stokes flow on all relevant spatial scales. We mitigate this problem by applying large eddy simulation (LES), which aims to model, rather than resolve, motion on scales below the filter length, which is fixed by a model parameter. By considering a periodic spatial domain, we avoid complications that arise in LES modelling in the presence of boundary layers. We consider the motion of an LES fluid subject to a constant body force of the Taylor-Green type as the separation between the forcing length scale and the filter length is increased. In particular, we discuss the transition from laminar to weakly turbulent motion, regulated by simple invariant solution, on a grid of $32^3$ points

A homoclinic tangle on the edge of shear turbulence

Experiments and simulations lend mounting evidence for the edge state hypothesis on subcritical transition to turbulence, which asserts that simple states of fluid motion mediate between laminar and turbulent shear flow as their stable manifolds separate the two in state space. In this Letter we describe a flow homoclinic to a time-periodic edge state. Its existence explains turbulent bursting through the classical Smale-Birkhoff theorem. During a burst, vortical structures and the associated energy dissipation are highly localized near the wall, in contrast to the familiar regeneration cycle

Time scale interaction in low-order climate models

Over the last decades, the study of climate variability has attracted ample attention. The observation of structural climatic change has led to questions about the causes and the mechanisms involved. The task to understand interactions in the complex climate system is particularly di±cult because of the lack of observational data, spanning a period of time typical for natural climate variability. One way around this problem is to represent the earth s climate in a computer model, as a set of prognostic equations. A disadvantage of this approach is that, if the model under consideration is to faithfully represent the climate system, it has to be large in terms of the number of degrees of freedom. This puts it out of reach of the ordinary analysis of dynamical systems theory. Alternatively, we can impose symmetries, consider limits of physical parameters, exploit perturbation theory and use Galerkin approximation to obtain simplified models of the earth s climate. Such models should highlight some isolated aspects of climate dynamics. A feature these simplified models have in common is the presence of widely different time scales. Throughout this thesis the emphasis is on the question to what extent the slow time scales play a role in the model s dynamics. The slow time scales are related to ocean dynamics and the fast time scales to atmospheric dynamics. The atmosphere model, studied here, was introduced by Edward Lorenz (1984). In chapter 2 a derivation of this model is given and it is shown that the Lorenz-84 model describes the jet stream in the mid-latitude atmosphere, and planetary waves, which can grow if the jet stream becomes dynamically unstable. The Lorenz-84 model is coupled to two different low-order ocean models. In chapter 3, it is coupled to Stommel s two box model. Stommel s model mimics the thermohaline circulation in the North Atlantic ocean. The typical time scale of variability of this circulation is of the order of centuries. This will be the longest time scale in the coupled models. In chapter 4, the Lorenz-84 model is coupled to an ocean model formulated by Leo Maas (1994). A physical description of the coupling is given. Apart from the overturning circulation, Maas model represents a wind driven gyre. There is coupling through exchange of heat at the surface and through wind shear forcing. The latter acts on a time scale of about one year, in between the fast atmospheric time scale and the slow overturning time scale. The simplified models are sets of coupled, nonlinear, ordinary differential equations. These can be analysed with the aid of dynamical systems theory. The emphasis will be on bifurcation analysis. Also, the time scale separation leads to the presence of small parameters in the equations. The consequences for the behaviour of the coupled models are explored by means of singular perturbation theory. In both coupled models, intermittent behaviour is observed. The slow subsystem, i.e. the ocean model, repeatedly pushes the fast subsystem, i.e. the atmosphere model, through a sequence of bifurcations. Thus, the ocean model plays an active role in the coupled system. Secondly, in the Lorenz-Maas model a periodic solution is shown to exist, with a period on the slow, overturning time scale. Along this solution the behaviour of the coupled model is dictated by internal ocean dynamics. Both these phenomena occur near a critical point of the coupled system, in agreement with the general idea that in climate models the slow components can play an active role near such critical points and are passive otherwise