Oscillons Near Hopf Bifurcations of Planar Reaction Diffusion Equations
McQuighan, Kelly Teresa (creator)
Sandstede, Bjorn (Director)
Menon, Govind (Reader)
Dafermos, Constantine (Reader)
Brown University. Applied Mathematics (sponsor)
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.