|
|
|
|
Home |
People |
Research |
Publications |
|
Meetings |
Jobs |
Contact |
Links | Search |
| Current BCANM seminars | |||||||||
Cavity solitons (CS) are self-localised structures in driven optical cavities containing a nonlinear medium, and could be natural "bits" for parallel processing of optical information. As dissipative spatial structures they share a number of fundamental properties with localised structures in other fields of physics, such as granular media, chemical reactions, gas discharges or fluids.
A mathematical description of the bifurcation sequence of a general class of localised structures including CS was recently given in Refs. [1,2]. Here, we analyse in detail the structure of the phase space on a specific nonlinear optical system: a ring cavity filled with a self-focusing Kerr medium. We find that the stable and unstable manifolds of homogeneous and pattern solutions present a much higher level of complexity than previously assumed, including the existence of additional fronts and localised solutions.
We also analyse the properties of asymmetric CS clusters (non-reversible trajectories). Such states are not, in general, homoclinic trajectories of a reversible dynamical system. We show, however, that asymmetric states can be moving solutions of the partial differential equation. The extra degree of freedom associated with their motion then allows such solutions to persist over a finite parameter range. The level of complexity is therefore increased with respect to that present in the ordinary differential equations describing the stationary solutions.
[1] P.D. Woods and A.R. Champneys, Physica D 129, 147 (1999).
[2] P. Coullet, C. Riera and C. Tresser, Phys. Rev. Lett. 84, 3069 (2000); Chaos 14,
193 (2004).
In 1956, M. King Hubbert made the prediction that overall oil production would peak in the US in the early 1970s, based on a simple model (logistic equation) for the oil production. The solution to this equation (logistic curve) is often referred to as the Hubbert Curve, which is symmetric in shape with a single maximum, the Hubbert Peak. While this model worked well for US oil production, there is serious doubt as to whether it also applies to world oil production, the peak of which might be imminent.
In this talk, an introduction is given to the Hubbert Peak, its fundamental principle and what the current estimates for its occurence are. Next, we will discuss higher-order Hubbert models, resembling supply-demand models (2D), or supply-demand-reserves models (3D). These nonlinear systems exhibit nonsymmetric production curves, where the Hubbert Peak is shifted into the future with a more rapid drop-off past the peak. In the 2D case, the model can be fully understood by phase-plane analysis. However, a more realistic model must inevitably include the economic aspect of world oil production, i.e. price and economic growth factors, from which researchers shy away.
This work was carried out with Stephen Korte during a summer project at UOIT with the financial support of NSERC, Canada
Spiral waves are planar patterns that have been observed in various biological, chemical and physical systems, as well as in numerical simulations of reaction-diffusion systems and complex Ginzburg-- Landau equations. Part of their fascination is due to the intriguing instabilities, such as meandering and drifting, core and far-field breakup, and spatio-temporal period doublings, that they exhibit. Among the challenges for theoretical studies of spirals is the task of relating these instabilities to spiral spectra. In this talk, I will present some recent results that help to predict discrete point eigenvalues that appear to be responsible for core and far-field breakup. This study was motivated by recent spectral computations by Wheeler and Barkley.
A coupled network of dynamical systems can exhibit a range of interesting behavior that may be qualitatively very different from their behavior in isolation. The so-called oscillator death is one such example, where limit-cycle oscillators stop oscillating when coupled and tend to a stable equilibrium instead. Furthermore, for a network of identical oscillators, such behavior is only possible in the presence of transmission delays. This talk will present a general characterization of oscillator death in the setting of coupled systems of functional differential equations whose oscillatory character results from a supercritical Hopf bifurcation. Using center manifold techniques and ideas from graph theory, the stability of the equilibrium solution is determined in terms of the delays and the spectrum of the graph Laplacian, for both directed or undirected networks. The results are also extended to discrete-time systems, and a complete characterization is given for conditions under which a network of coupled (and possibly chaotic) maps can be driven to a fixed point by an appropriate choice of delays.
In this talk we start with a short review of the well known and widely-used one-layer evolution equation of the free surface in thin liquid films. Generally this equation can be written as \[ \partial_t h(x,y,t) = -\nabla \vec S(h(x,y,t),x,y)\nonumber \] where $h$ denotes the surface height and the r.h.s is the divergence of a flux $\vec S$. Here, {\it two-layer} systems of immiscible fluids are investigated and two different possible configurations are shortly discussed. After this introductory part we derive systematically a single evolution equation for the interface of vertically bounded two-layer systems in long-wave approximation. This derivation starts from the basic hydrodynamic equations (incompressible Navier-Stokes-equations) and is based on long-wave scalings of the system. The resulting evolution equation is given in a general form and allows to incorporate any effects easily in the corresponding ``long-wave'' order. Appropriate limits recover the widely-used one-layer systems. Furthermore new characteristics of our evolution equation are shortly discussed. In the following we focus mainly on gravity and thermocapillarity (Marangoni effect). A linear stability analysis reveals characteristic long-wave features as well as new two-layer effects.
Additionally, such ``simple'' forces allow to define a Lyapunov functional and consequently to predict the long-time solution without numerical integration of the fully nonlinear equation. Moreover, the free energy density reveals possible metastable states. Numerical simulations of the fully nonlinear equation verify all former statements. Furthermore, numerical experiments allow: 1) to visualize pattern formation processes like drop or hole formation and 2) to distinguish linear short-time and nonlinear long-time regimes. In the (strongly nonlinear) long-time domain coarsening coefficients can be extracted which reveal in turn new characteristic features of two-layer systems.
A vertically vibrated layer of shear-thickening fluid will form localized structures. In my experiments, a mixture of cornflour/water or glass beads/water is vibrated at frequencies up to 200 Hz and accelerations up to 30 g in a 10 cm wide container with a layer of depth 5-10 mm thick. Above a critical acceleration (approximately 10 g) a perturbation of the flat surface will grow until it reaches the bottom of the container, forming a circular hole. These holes are typically on the order of 10 mm in diameter, and they can remain stable for more than 10^5 periods. At higher accelerations the rim of the hole becomes unstable; either a tongue grows upward from the rim and eventually collapses onto the hole annihilating it or the hole divides into two separate ones. At even higher accelerations, the entire layer writhes in a disordered manner. The mechanism for these instabilities is unknown. I will present experimental correlations between these instabilities and the fluid's rheological proprieties.
We propose a method for control of synchronization in ensembles of interacting oscillators. We suggest to use a nonlinear delayed feedback, where a synchronized population of oscillators is stimulated with a signal constructed by using the delayed mean field of the ensemble nonlinearly combined with its instantaneous mean field. The stimulation results in complete desynchronization of the oscillators and restores their natural frequencies, so that the oscillators rotate as if they were uncoupled. Moreover, the amplitude of the stimulation signal practically vanishes when a desynchronized state is achieved. Some other properties of the impact of nonlinear delayed feedback on stimulated ensembles of coupled oscillators, like multistability of stimulation-induced desynchronized states and sensitivity to frequency control are discussed. We suggest our method for mild and effective deep brain stimulation in neurological diseases characterized by pathological cerebral synchronization.
The talk will concentrate on the various noise and vibration phenomena that are of current concern to the power train engineers, particularly in the automotive industry. The particular class of problem relates to impact-induced noise and vibration, either under repetitive (cyclic) conditions or due to sudden changes in system dynamics (impulsive). All these phenomena are progressively regarded as quality issues by customers in the automotive industry. The seminar will provide some insight to the various phenomena of concern, and the findings of investigations carried out by vehicle and rig-base tests, as well as through numerical analysis.
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.
