Speaker: Snezhana Abarzhi, U. Chicago
Title: The effect of heterogeneities and noise on pattern formation and chaos development

Abstract: Pattern formation occurs in a broad variety of natural and artificial systems and exhibits some universal features of the nonlinear dynamics. This universality is well captured by low-dimensional models that study order and chaos in ideal spatially extended systems. Realistic nonlinear systems are multi-scale and inhomogeneous. An important issue to consider is the qualitative and quantitative correspondence between the idealized low-dimensional description and a realistic, heterogeneous and multi-scale, phenomenon. This can be viewed on one hand as a sensitivity of modeling results to heterogeneities and noise, and, on the other hand, -- as a control of a pattern forming phenomenon by means of heterogeneities. The heterogeneities induce spatial and temporal modulations (either regular or random), and for accurate capturing this influence, the modulations should be accounted for in a 'non-intrusive' way that preserves internal symmetries of the system. We consider the effect of modulations on the non-equilibrium dynamics of wave patterns in the framework of a complex Ginzburg-Landau equation (CGLE) with parametric non-resonant forcing that preserves the gauge invariance of the system. The forcing results in occurrence of traveling waves with new dispersion properties as well as to monochromatic wave with quasi-periodic and (in case of large-scale forcing) essentially anharmonic spatial structures. Furthermore, the forcing may completely suppress the development of an intermittent chaos. We discuss the connection of our model to pattern-forming systems described by real Ginzburg-Landau and nonlinear Schrodinger equations, as well as applications of the model in convection, dynamics of atmosphere and ocean, rheology and optics.

- S. I. Abarzhi, O. Desjardins, A. Nepomnyashchy, and H. Pitsch (2007), Influence of parametric forcing on the nonequilibrium dynamics of wave patterns, Phys. Rev. E 75, 046208
- A. Nepomnyashchy and S. I. Abarzhi (2010), Monochromatic waves induced by large-scale parametric forcing, Phys. Rev. E 81, 037202


Time: Friday, April 30, 2010, 12:30-1:30 p.m.

Place: Science and Tech I, Room 242

Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, VA 22030-4444
http://math.gmu.edu/
Tel. 703-993-1460, Fax. 703-993-1491