Pattern formation

Pattern formation is an ubiquitous phenomenon in the dynamics of extended nonlinear systems [CH93,GL99]. Patterns in extended systems arise as a result of the interplay of many factors including nonlinearities, external forcing and/or excitability of the medium, spatial interactions, and internal dissipation.

Continuum Coupled maps

Motivated by experiments on vibrated granular layers by Paul Umbanhowar and coworkers at the Center for Nonlinear dynamics, we recently introduced Continuum Coupled Maps, a new framework for studying pattern formation in periodically forced systems.

A continuum coupled map (CCM) is a dynamical system defined on the space of smooth functions $\xi$ : D $\rightarrow$ $\mathbb {R}$ by

$$\xi_{{n+1}}^{} \xi_{n}^{}$$ (CCM)

where D is a continuous spatial region, M is a nonlinear map, G is a smooth kernel, o denotes functional composition and * denotes convolution. M and G introduce nonlinearity and spatial coupling respectively. This framework therefore allows one to model a variety of extended systems through appropriate choices for G and M.

The system (CCM) is invariant under translations in space as well as time translations by one unit. For this reason, the CCM framework is useful in modeling periodically forced extended systems which have the same symmetries. This framework is also very convenient for numerical simulations.

Vibrated granular layers

We used the Continuum coupled map approach to successfully model recent experiments at U. Texas, on pattern formation in vibrated granular layers [MUS95,UMS96,UMS98]. Using a generic model where Temporal period doubling interacts with pattern formation at a preferred scale, we obtain the spatio-temporal bifurcation phenomena observed in experiment.

The figure shows the phase diagram that we obtain from our model. Our model is a generic model for pattern forming systems that have period doubling and is similar in spirit to the Swift-Hohenberg and the Complex Ginzburg-Landau Equations. It is not tied to the physics of any specific experiment and the features that it reproduces can be thought of as being universal in systems with interacting temporal period doubling and spatial pattern formation.

Extended patterns in our model. (a) Period 2 Stripes. (b) Period 2 Squares (c) Period 2 Hexagons (d) Period 2 flat state with a front. (e) Period 4 stripes (f) Disorder.

Localized patterns in our model. (a) Two oscillons in opposite temporal phases. (b) A bound state of oscillons with coordination number 3

Fronts in periodically forced systems

Using the CCM approach with a map that has a coexisitng stable periodic orbit and a chaotic attractor, we investigated the propagation of fronts between domains of spatio-temporal chaos and homogeneous steady states [KVO⁺01]. We obtain effective equations for the front and also scaling laws for its roughness using an effective long-wavelength model. These laws are verified by direct numerical simulations on our model. In particular, the chaos plays the same role in the roughening of the front, as noise in the Kardar-Parisi-Zhang equation []. Also, the ``averaged'' front forms finite time singularities (cusps) consistent with predictions of the long wavelength model.

Multiple Scale Behavior, Coarsening and Domain Growth

Pattern formation is intrinsically a multiple scale phenomenon, since it results from an interplay of many different physical processes, each acting on different spatial/temporal scales. In our model for vibrated granular layers [VO98,VO01], we take D = [0, L] x [0, L], the initial function $\xi_{0}^{}$(x) = $\bar{{\xi}}_{0}^{}$ + $\delta$$\eta$(x), where $\bar{{\xi}}_{0}^{}$ is an order 1 constant, $\delta$ $\ll$ 1 is small, and $\eta$ is a normalized random function. By evolving this function according to (CCM) we observe the following:

1. Given a point x₀ $\in$ D, the action of the nonlinear map M typically produces changes in the value $\xi_{n}^{}$(x₀) on every iteration.
2. $\lambda$ is the range of G, i.e., G(x) decays rapidly for |x| $\gg$ $\lambda$. The system displays coherent behavior at this length scale on a time scale $\tau$, leading to the onset of patterns of wavelength $\sim$ $\lambda$. Due to the short range of the spatial coupling G, the patterns are not coherent across the entire region D.
3. For times t $\gg$ $\tau$, r(t) is the typical size of a domain on which the pattern is coherent. The domains coarsen and a single domain fills the entire region on a time scale T.

The system has multiple relevant time scales 1 $\ll$ $\tau$ $\ll$ T, and length scales a $\ll$ $\lambda$ $\ll$ r(t) $\lesssim$ L, where a is a small scale cutoff. The analysis of (CCM) is complicated by this multiple scale behavior. By analyzing a system obtained by carefully truncating (CCM), we solved for the long time asymptotics of (CCM), thereby solving the pattern selection problem in our model for vibrated granular layers [VO01]. We are currently using this approach for other extended, periodically forced systems.

We are currently looking at the following question -

Problem 8   Determine the intermediate asymptotics for (CCM), i.e. find the appropriate equations for the motion of the domain walls in the coarsening process.

On the typical scale of a domain r(t), the coherent pattern with wavelength $\lambda$ is the microstructure. We will investigate the relation between the macroscopic motion of the domains and the details of the patterns inside the domains (the microstructure) which we will characterize by Young measures or H-measures [Tar92] generated by $\xi_{n}^{}$(x/r(t)) as $\lambda$/r(t) $\rightarrow$ 0.