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Analysis of delayed bifurcations in reaction-diffusion systems

Banff International Research Station for Mathematical Innovation and Discovery

We analyze examples of delayed bifurcations in reaction-diffusion systems in both the weakly and fully nonlinear regimes. The delay effect results as the system passes slowly from a stable to unstable regime, and was previously analyzed in ODEs in [P.Mandel and T.Erneux, J.Stat.Phys 48(5-6) pp.1059-1070, 1987]. For spike solutions in the fully nonlinear regime, we demonstrate that delay can be quantified for a special class of problems in which the linear stability problem is explicitly solvable. In the weakly nonlinear regime, in the context of a simplified Klausmeier model for vegetation patterns, we analyze how addition of random noise can affect the magnitude of delay. In both regimes, we show that delay can play a critical role in determining the eventual fate of the system. Joint works with Yuxin Chen, Chunyi Gai, Theodore Kolokonikov, and Michael Ward.

Mathematics; Partial differential equations; Dynamical systems and ergodic theory

video/mp4

Author affiliation: University of British Columbia

2017-01-31

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60085

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/