We study the existence of multiple equilibrium states in a simple fluid network using Newtonian fluids and laminar flow. We demonstrate theoretically the presence of hysteresis and bistability, and we confirm these predictions in an experiment using two miscible fluids of different viscosity—sucrose solution and water. Possible applications include blood flow, microfluidics, and other network flows governed by similar principles.

We consider pulse formation dynamics in an actively mode-locked laser. We show that an amplitude-modulated laser is subject to large transient growth and we demonstrate that at threshold the transient growth is precisely the Petermann excess noise factor for a laser governed by a nonnormal operator. We also demonstrate an exact reduction from the governing PDEs to a low-dimensional system of ODEs for the parameters of an evolving pulse. A linearized version of these equations allows us to find analytical expressions for the transient growth below threshold. We also show that the nonlinear system collapses onto an appropriate fixed point, and thus in the absence of noise the ground-mode laser pulse is stable. We demonstrate numerically that, in the presence of a continuous noise source, however, the laser destabilizes and pulses are repeatedly created and annihilated.

We explore the stability of equilibrium solution(s) of a simple model of microvascular blood flow in a two-node network. The model takes the form of convection equations for red blood cell concentration, and contains two important rheological effects—the Fåhræus–Lindqvist effect, which governs viscosity of blood flow in a single vessel, and the plasma skimming effect, which describes the separation of red blood cells at diverging nodes. We show that stability is governed by a linear system of integral equations, and we study the roots of the associated characteristic equation in detail. We demonstrate using a combination of analytical and numerical techniques that it is the relative strength of the Fåhræus–Lindqvist effect and the plasma skimming effect which determines the existence of a set of network parameter values which lead to a Hopf bifurcation of the equilibrium solution. We confirm these predictions with direct numerical simulation and suggest several areas for future research and application.

We study theoretically the effect of transverse boundary conditions on the traveling waves found in infinitely extended and positively detuned laser systems. We find that for large-aspect-ratio systems, well above threshold and away from the boundaries, the traveling waves persist. Source and sink defects are observed on the boundaries, and in very-large-aspect-ratio systems these defects can also exist away from the boundaries. The transverse size of the sink defect, relative to the size of the transverse domain, is important in determining the final pattern observed, and so, close to threshold, standing waves are always observed.

Several experimental groups have demonstrated communication with chaotic lasers. We analyze data collected from a message-modulated erbium-doped fiber-ring laser (provided by VanWiggeren and Roy). We show that the transmitted signal is dominated by convolution of the message with the response function of the laser. A simple model based on the topology of the laser allows us to recover a hidden message. While prior estimates indicate that the laser dynamics are high dimensional, we show that only four parameters are required, each of which can be recovered from the transmitted signal alone.

Pattern formation is analyzed and simulated in a nonlinear optical system involving all three space dimensions as well as time in an essential way. This system, counterpropagation in a Kerr medium, is shown to lose stability, for sufficient pump intensity, to a nonuniform spatial pattern. We observe hexagonal patterns in a self-focusing medium, and squares in a self-defocusing one, in good agreement with analysis based on symmetry and asymptotic expansions.