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An Isaac Newton Institute Workshop

Developments in Experimental Pattern Formation

Snaking and Its Consequences

9th August 2005

Author: Edgar Knobloch (University of California at Berkeley)

Abstract

Recent simulations^1 of binary fluid convection in low Prandtl number liquids reveal the presence of multiple numerically stable spatially localized steady states at supercritical Rayleigh numbers. The length of these states decreases as the Rayleigh number decreases; below a critical Rayleigh number the steady states are replaced by relaxation oscillations in which the steady state is gradually eroded until no rolls are present (the slow phase), whereupon a new steady state regrows from small amplitude (the fast phase) and the process repeats. The Swift-Hohenberg equation (both variational and nonvariational) provides much insight into this behavior. This equation contains several classes of localized steady states whose length grows in a characteristic 'snaking' fashion as they approach spatially periodic states, and the associated dynamics resemble the binary fluid simulations. The origin of the snaking and the stability properties of the associated states will be elucidated, and the results used to shed light on the remarkable complexity of these simple systems.

This talk is based on joint work with Oriol Batiste and John Burke.

^1 O. Batiste and E. Knobloch, Phys Fluids 17,064102 (2005).