Oscillons Near Hopf Bifurcations of Planar Reaction Diffusion Equations

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Overview

Title
Oscillons Near Hopf Bifurcations of Planar Reaction Diffusion Equations
Contributors
McQuighan, Kelly Teresa (creator)
Sandstede, Bjorn (Director)
Menon, Govind (Reader)
Dafermos, Constantine (Reader)
Brown University. Applied Mathematics (sponsor)
Copyright Date
2014
Abstract
Oscillons are spatially localized, time-periodic structures that have been observed in many natural processes, often under temporally periodic forcing. Near Hopf bifurcations, such systems can be formally reduced to a forced complex Ginzburg--Landau (CGL) equation, with oscillons then corresponding to stationary localized patterns. In this thesis, stationary localized structures of the planar 2:1 forced CGL are investigated analytically and numerically. Four distinct localized solutions to the steady-state planar forced CGL are studied: localized offsets from both the trivial background state and a nontrivial background state, in each case with both monotone tails and oscillatory tails. The existence of both monotone solutions is proved analytically in regions where two spatial eigenvalues associated with the appropriate linearization of the one-dimensional CGL collide at zero. The numerical study complements the analytical results away from onset. The numerical study also investigates the existence of localized solutions with oscillatory tails. One particular outcome is that planar oscillons with oscillatory tails can exist in parameter regions where one dimensional oscillons cannot.
Keywords
dynamical systems
pattern formation
oscillons
spatial dynamics
localized radial structures
geometric blow-up
complex Ginzburg--Landau equation
Dynamics
Notes
Thesis (Ph.D. -- Brown University (2014)
Extent
xv, 248 p.

Citation

McQuighan, Kelly Teresa, "Oscillons Near Hopf Bifurcations of Planar Reaction Diffusion Equations" (2014). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:386200/

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