|
|
|
|
Home |
People |
Research |
Publications |
|
Meetings |
Jobs |
Contact |
Links | Search |
| Current BCANM seminars | |||||||||
Friday 24 March 2006
The study of the transition from order to chaos is one of the central strands in Nonlinear Science. In quasiperiodically forced dynamical systems, the order is geometrically interpreted as existence of invariant tori carrying quasiperiodic motion, and the transition to chaos corresponds to the destruction of invariant tori. The seminal work of Grebogi, Ott, Pelikan and Yorke in 1984 showed the existence of strange non chaotic attractors (SNA) in dissipative systems, a mixture of regular dynamics (negative Lyapunov exponents) and strange dynamics (geometrically complicated attractors). Since then, there has been a lot of numerical work and some rigorous analysis. Armed with theoretical and numerical tools, we will review some of the examples that have appeared in the literature. The results reported have been obtained in collaboration with Rafael de la Llave, Joaquim Puig and Carles Simo.
Traffic flow on freeways is a nonlinear, many-particle phenomenon, with complex interactions between vehicles such as traffic jams, stop-and-go-waves. This talk presents a stochastic macroscopic model of freeway traffic suitable for on-line estimation, routing and ramp metering control. The freeway is considered as a network of interconnected components. The compositional model proposed here extends the Daganzo cell transmission model by defining sending and receiving functions explicitly as random variables, and by also specifying the dynamics of the average speed in each cell (segment). Stochastic equations describing the macroscopic traffic behaviour of each cell, as well as its interaction with neighbouring cells are obtained.
Next, we apply this model to estimation of traffic state in freeway networks. We show how the traffic estimation problem can be solved within Bayesian framework. We develop a particle filter and an Unscented Kalman filter. Particle filters are appropriate for traffic state estimation because they can deal with complicated and highly nonlinear models, with multi-modal posterior distributions and non-Gaussian signals. Measurements (from cameras or magnetic loops) are received only at boundaries between some segments and averaged within regular or irregular time intervals. This limits the measurement update in the PF and UKF to only these time instants when a new measurement arrives, with possibly many state updates in between consecutive measurement updates. The filters performance is validated and evaluated over synthetic and real traffic data from a Belgian freeway.
Knots and links formed by unstable periodic orbits in a strange attractor carry signatures of the stretching and squeezing mechanisms that organize it. We will review how this property has been harnessed to design a robust method for analyzing experimental chaotic data. In particular, we will present recent experiments in a nonstationary system where the knot type of an orbit has been used as an unambiguous signature of chaos. Possible extensions to higher dimensions will also be discussed.
The spike-diffuse-spike (SDS) model is an idealised model of dendritic tissue where excitable dendritic spines are connected to a passive dendritic tree. Spine-head dynamics are modeled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. We develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded on a simple one-dimensional cable. This system supports saltatory waves as a result of the discrete distribution of spines. In response to a periodic pulse train the SDS model shows a positive correlation between spine density and low-pass temporal filtering, as has been seen in the experimental studies. Moreover, the observed wave properties are robust to natural sources of noise that arise both in the cable and the spine-head. Finally, we demonstrate that the SDS framework can be generalised for branched dendritic structures.
We study a globally coupled map system, namely, stability and asymptotic behaviour of its solutions. Main attention is paid to the partial synchronization (or clustering) phenomenon, when all phase coordinates split into several groups of elements with identical dynamics, called clusters. Those solutions belong (or asymptotically tend) to a certain invariant linear subspace - cluster manifold. A keystone of the research is an analytic relation between longitudinal and transverse Lyapunov exponents obtained for such solutions. Due to this formula, sufficient conditions for transverse and overall stability were found. In addition, we make a brief analysis for a special subclass of partially synchronized solutions called quasi-clusters.
When a drop falls from a faucet, the Navier-Stokes equation forms a singularity at the point of separation. Singularities are vital to many engineering problems such as printing, painting, coating, and air entrainment where the fine details of how a drop pinches off or how a drop spreads are essential to controlling the process. The focusing inherent in a singularity can be used to make small things, as for example nanometre-sized fluid jets. This talk expounds the crucial concepts of universality, scaling, and matching needed to understand singularities. Universality means that singularities are independent of initial conditions, and thus organise the flow. Self-similar scaling often permits to describe singularities analytically. Finally, matching conditions relate the singularity to the large scale flow, or point to new micro-physics.
We investigate the structuring process of two-layer films driven by destabilizing van der Waals interactions. Using coupled film thickness evolution equations in three dimensions different types of dynamical morphological transitions are described. They occur in the course of the short- and the long-time evolution. We illustrate that a measure for an integral mode type faithfully captures transitions of the interface morphologies. Using an Si/PMMA/PS/air system as example transitions via branch switching and via coarsening are analysed.
Scientists attempt to understand physical phenomena by constructing models. A model serves as a link between scientists and nature, and one fundamental goal is to develop models whose solutions accurately reflect the nature of the physical process. A dynamical model uses simplifying assumptions and approximations in the hope of capturing the essential characteristics of how a physical system evolves with time. The question of whether a model accurately reflects nature is one constantly faced by scientists. Recently we have discovered that there exists a new level of mathematical difficulty, brought from the theory of dynamical systems, which can limit our ability to represent nature using deterministic models. Specifically, we have discovered that certain chaotic systems found in nature cannot be modelled, particularly higher-dimensional chaotic systems. No model of such a system produces solutions of reasonable length that are realized by nature. (Furthermore, for these processes, the numerical solutions of the models do not approximate any actual model solutions.)
