Snakes propel themselves over land using a variety of techniques, including sidewinding, lateral sinuous slithering and a unidirectional accordion-like mode. We explore these friction-based propulsion mechanisms through a combined experimental and theoretical investigation. Particular attention is given to classifying the gaits of snakes according to Froude number and the relative magnitudes of the frictional forces in the tangential and normal directions. In a simple kinematic model, we prescribe the waveform of the snakeand calculate its motion as required by the torque and force balances on its body. A key feature of our model is that it allows us to rationalize the snake'sgait on the basis of speed and mechanical efficiency.

The complexity and unwieldiness of computational fluid dynamics software form a substantial barrier to entry to numerical research in turbulence. Channelflow is an effort to make numerical analysis easy for a particular kind of flow (channel and plane Couette flow). Channelflow provides (1) a high-level, flexible C++ class library in which key algorithms can be expressed compactly, and (2) implementations of those algorithms that can be run from the Unix command line. In this talk I'll outline the numerical problems, talk about what makes computational fluid dynamics hard, and provide an overview of the Channelflow "language", its command-line utilities, solution database, installation, limitations, and future plans.

Turbulence is usually characterized by scaling, energy cascade and intermittency, concepts which are presented in the concepts of complex systems not usually thought of as being related to fluid dynamics. This talk discusses whether or not turbulence is ever fully-developed at a finite Reynolds number, by analogy with recent work on anomalous scaling in the (mean velocity)-(distance from wall) relationship found in pipe flow. I discuss recent experimental observations that the probability distribution of power fluctuations has a universal form, apparently indistinguishable from that seen in the 2D XY model well below the Kosterlitz-Thouless transition. Lastly, I present evidence from a reanalysis of turbulent pipe flow in smooth and rough pipes for a complete analogy between turbulence and critical phenomena, going beyond power-law scaling in spectral structure.

In shear flows, the transition to turbulence typically occurs through a subcritical bifurcation where a finite amplitude perturbation is required to take the system from the laminar state to a turbulent one. Experiments have shown that the lifetime of the turbulent state is finite at moderate Reynolds numbers. Some experiments suggest that above some critical Reynolds number turbulence becomes sustained, whereas other experiments suggest its lifetime increases rapidly but diverges only in the limit of infinite Reynolds number. I will briefly discuss the history of the problem, the consequences of each result, what they mean in the context of dynamical systems, and how they affect our prospects for controlling turbulence. I will then present the first (to my knowledge) measurements of the turbulent state lifetimes in the flow between concentric, rotating cylinders in the regime where the transition to turbulence is subcritical. The streamwise periodicity of this flow allows for arbitrarily long observation times, which has allowed us to observe turbulent decays over several orders of magnitude in lifetimes. Because the lifetimes of turbulent transients have never been measured in this geometry, our study also allows us to test whether the transient nature of the turbulence observed in previous experiments is specific to those flow geometries or is present in a more general class of shear flows. Finally, I will discuss some open questions and what is being done to answer them.

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.

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.

Recent experiments report that multimode fiber lasers can be coherently combined without any active control. This spontaneous formation of coherent state is well explained by our iterative map model. I will discuss the dynamics of the phenomena of coherent emission from the lowest-loss output facet and see how bifurcation analysis can unfold the problem. Limit of coherent addition of lasers is also discussed based on the idea of lowest-loss mode selection.

In the financial world, one has to deal with randomly fluctuating prices of different "products" like a stock, for instance. How does one deal with the randomness in financial markets? What guiding principles can one use in pricing financial contracts in the presence of such randomness? This talk will take a heuristic approach to introduce the basic guiding principles of financial derivative (a financial contract) pricing, and touch upon simple results when the stock price can be treated as a diffusion process.

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. Our goal is to determine the finest possible partition of the state space for a given hyperbolic dynamical system and a given weak additive white noise of specified strength. We test these ideas on the ``skew Ulam'' repeller, by computing the local eigenfunctions of the Fokker-Planck evolution operator in the neighborhood of each periodic point, and use their widths to attain an optimal partition of the state space. The Fokker-Planck evolution is then represented by a finite Markov graph, whose determinant is used to evaluate the escape rate of the repeller.

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.