It has recently become possible to compute precise equilibrium, traveling wave, and periodic orbit solutions to pipe and plane Couette flow at moderate Reynolds numbers. These invariant solutions (1) capture the complex dynamics of rolls and streaks (coherent structures) in wall-bounded flows and (2) provide a framework for understanding turbulent flows as dynamical systems. We present a number of weakly unstable equilibria, traveling waves, and periodic orbits of plane Couette flow and a new method of visualizing their state-space dynamics. What emerges is a picture of low-Reynolds turbulence as a walk among a set of weakly unstable invariant solutions.

I use the Navier-Stokes equations for barotropic turbulence as a zero-order approximation of chaotic space-time patterns and equilibrium distributions that mimic turbulence in geophysical flows. In this overly-simplified set-up for which smooth-solutions exist, we investigate if is possible to bound the uncertainty associated with the numerical domain discretization, i.e. with the limitation imposed by the Reynolds number range we can explore. To do so I analyze a series of stationary barotropic turbulence simulations spanning a range of Reynolds number of 105 and run over a three year period for over 300,000 CPU hours. A persistent Reynolds number dependency in the energy power spectra and second order vorticity structure function is found, while distributions of dynamical quantities such as velocity, vorticity, dissipation rates and others are invariant in shape and have variances scaling with the viscosity coefficient according to simple power-laws. The relevance to this work to climate models will be discussed.

Recent experiments using passively coupled fiber lasers report coherent emission even though members of the laser array are poorly matched in length and operate on thousands of longitudinal modes. I discuss our latest theoretical and experimental contributions to understanding this phenomenon and extending it to larger arrays. Our model predicts - and our experiments confirm - a simple pathway to achieving a robust, synchronized output.

The Helium atom driven by an external pulse exhibits single and double ionizations. Some of the double ionized electrons result from sequential double ionization, others correspond to non-sequential double ionization. The double ionization probability takes the form of a ''knee'' as a function of the intensity of the pulse. Here, we investigate the classical dynamics of this system. Using Lyapunov indicators, periodic orbits, as well as reduced Hamiltonian models, we investigate the mechanisms occurring in phase space which shed light on the single and double ionization mechanisms. We find that although the Helium in the absence of the laser field is extremely chaotic, the mechanisms behind all three kinds of ionization are based on integrable models and explain the ''knee'' observed in experiments and numerical simulations.

The Hamiltonian description of the self-consistent interaction between an electromagnetic plane-wave and a co-propagating beam of charged particles is considered. We show how the motion can be reduced to a one-dimensional Hamiltonian model (in a canonical setting) from the Vlasov-Maxwell Poisson brackets. The reduction to this paradigmatic Hamiltonian model is performed using a Lie algebraic formalism which allows us to remain Hamiltonian at each step of the derivation.

We will consider the dynamics of passive tracers in a system of three point vortices and in an array of vortices. These dynamics are all particular examples of chaotic advection. This phenomenon has fundamental implication in various physical systems. On large scales one may for instance think as plasma physics (confinement), geophysical flows (pollution). But also on a much smaller scale for micro-fluidic devices, for which chaotic advection seems to be the best candidate in order to trigger mixing wihtout breaking anything. For the point vortex flow some motivation of the choice of the flow and the dynamics of vortices will be given. Then transport properties of the tracers will be discussed and the anomalous effects induced by the stickiness around regular islands presented. While for the array of vortices I will present a way to perturb the flow in order to enhence mixing while limiting transport.