Punit R. Gandhi

We have designed an introductory laboratory course that engaged first-year undergraduate students in two complementary types of iteration: (1) iterative improvement of experiments through cycles of modeling systems, designing experiments, analyzing data, and refining models and designs; and (2) iterative improvement of self through cycles of reflecting on progress, soliciting feedback, and implementing changes to study habits and habits of mind. The course consisted of three major activities: a thermal expansion activity, which spanned the first half of the semester; final research projects, which spanned the second half of the semester; and guided student reflections, which took place throughout the duration of the course. We describe our curricular designs and report examples of student work that demonstrate students' iterative improvements in multiple contexts.

The Adler equation with time-periodic frequency modulation is studied. A series of resonances between the period of the frequency modulation and the time scale for the generation of a phase slip is identified. The resulting parameter space structure is determined using a combination of numerical continuation, time simulations and asymptotic methods. Regions with an integer number of phase slips per period are separated by regions with noninteger numbers of phase slips, and include canard trajectories that drift along unstable equilibria. Both high and low frequency modulation is considered. An adiabatic description of the low frequency modulation regime is found to be accurate over a large range of modulation periods.

The theory of stationary spatially localized patterns in dissipative systems driven by time-independent forcing is well developed. With time-periodic forcing related but time-dependent structures may result. These may consist of breathing localized patterns, or states that grow for part of the cycle via nucleation of new wavelengths of the pattern followed by wavelength annihilation during the remainder of the cycle. These two competing processes lead to a complex phase diagram whose structure is a consequence of a series of resonances between the nucleation time and the forcing period. The resulting diagram is computed for the periodically forced quadratic-cubic Swift--Hohenberg equation, and its details interpreted in terms of the properties of the depinning transition for the fronts bounding the localized state on either side. The results are expected to shed light on localized states in a large variety of periodically driven systems.

The generalized Swift--Hohenberg equation with a quadratic-cubic nonlinearity is used to study the persistence and decay of localized patterns in the presence of time-periodic parametric forcing. A novel resonance phenomenon between the forcing period and the time required to nucleate one wavelength of the pattern outside the pinning region is identified. The resonance generates distinct regions in parameter space characterized by the net number of wavelengths gained or lost in one forcing cycle. These regions are well described by an asymptotic theory based on the wavelength nucleation/annihilation time near the boundaries of the pinning region. The resulting theory leads to predictions that are qualitatively correct and, in some cases, provide quantitative agreement with numerical simulations.