Gabriela Jaramillo

RESEARCH

Nonlocal Effects in Oscillatory Media

Oscillatory media is known for creating beautiful patterns like spiral, target, and traveling waves. At the heart of these systems there are two mechanisms whose interaction contributes to the formation of these structures. One is the intrinsic dynamics of the system, which leads to oscillations and justifies the name oscillatory media. The second mechanism is the interaction, or coupling, that takes place between these `oscillators'. In many cases this coupling is well approximated by a local diffusion process, but when nonlocal interactions become important they can lead to novel structures. My current research in this area focuses on rigorously deriving amplitude equations for these systems, showing existence of target patterns and spiral waves, and understanding how these solutions can transform into patterns with regions of coherence and incoherence (chimera states).

Defects

I study the effect of defects and heterogeneities in pattern forming systems. In particular, I want to understand how these impurities alter or create patterns in oscillatory media (figures a) and b) ) and in convection phenomena (figure c) ). In these examples defects are localized areas were the properties of the medium are different. As a result these impurities can be modeled as perturbations of the amplitude equations that describe these systems. The main difficulty with this approach is that the linearization about the steady states is in general not a invertible operator in regular Sobolev spaces. My research has focused on developing methods to make these operators Fredholm so that regular perturbation techniques can be employed to find solutions.

a)

b)

c)



Ribbons

Thin strips of paper or acetate when subjected to twist and tension exhibit a wide range of geometric configurations. Depending on parameters (tension, twist, length, and thickness) the experiments show creased, wrinkled or plain helicoids, as well as loops, cylinders, and geometries with self-contact zones. One of the main focus in this area is to understand the instabilities that lead to these different configurations. Recently, experiments have shown that the material transverse stresses play an important role in the transition from wrinkled helicoids to loops. My goal is to understand the effects of these stresses and how they contribute to the ribbon's shape transitions. To do this, together with Dr. Venkataramani, we are developing a numerical scheme to find minimizers of an energy based on the Foppl von-Karaman's plate equations. Our scheme is an adaptation of the Split-Bregman iteration, (a widely used algorithm in compressed sensing and image de-noising applications) to problems where one seeks minimizers of a non-convex functional.

PUBLICATIONS