We report the results of a study to model the behavior of nonlinear metamaterials in the microwave frequency range
composed of arrays of split-ring resonators combined with nonlinear circuit elements. The overall model consists of an
array of coupled damped oscillators whose inter-element coupling is a function of signal amplitude, similar to that which
exists in the Fermi-Pasta-Ulam system.  We note the potential occurrence of classical nonlinear effects including
parametric coupling, FPU recurrence and chaos. These effects lead to nonlinear waves on the array which are a type of
soliton particular to the form of nonlinearity that has been incorporated. We have studied, in particular, the nonlinear
effects that arise from tunnel diodes embedded in the resonant circuits. We carry out simulations of the resulting circuit
The nonlinear electromagnetic response is one of the foundations of modern technology and it arises in natural
materials at the atomic scale. We briefly present some of the fundamentals of nonlinearity in natural materials
and then we present experimental studies of analogous behavior in meta-atoms, the fundamental building block
of metamaterials. Specifically tunnel-diode loaded, microwave split-ring resonators are shown to enable various
nonlinear phenomena including self-sustained oscillation, harmonic/comb generation, frequency locking/pulling,
and quasi-chaos generation. We discuss the possible adaptation of these unit cells to create bulk nonlinear metamaterials.
Next generation x-ray sources require very high-brightness electron beams that are typically at or beyond the present state-of-the-art, and thus place stringent and demanding requirements upon the electron injector parameters. No one electron source concept is suitable for all the diverse applications envisaged, which have operating characteristics ranging from high-average-current, quasi-CW, to high-peak-current, single-pulse electron beams. Advanced Energy Systems, in collaboration with various partners, is developing several electron injector concepts for these x-ray source applications. The performance and design characteristics of five specific RF injectors, spanning "L" to "X"-band, normal-conducting to superconducting, and low repetition rate to CW, which are presently in various stages of design, construction or testing, is described. We also discuss the status and schedule of each with respect to testing.