Observation of Front Bifurcations in Controlled Geometries: From One to Two Dimensions

Order parameter equations for front transitions: Planar and circular fronts

“…The multivalued relations correctly predict spontaneous front transitions induced by curvature [20,25], but cannot describe dynamics during front transitions. Differential order parameter equations, like (30), give a more accurate characterization of the dynamics. These differential equations reduce to an algebraic C −κ relation when the time scale of front transitions becomes much shorter than the time scale of curvature changes.…”

“…The coupled sets of equations (25) and (30) exhibit Hopf bifurcations from stationary to oscillating fronts (breathing spots). Equations (30) exhibit two additional behaviors pertaining to rebounding and collapsing spots in the full equations.…”

“…Figure 4 shows three different possibilities for the dynamics of circular fronts. The thick trajectories represent dynamics computed by numerical solution of the coupled equations (30). The initial conditions correspond to a large shrinking up state spot.…”

“…Models of this type have been studied in various physical and chemical contexts [1,2,[29][30][31][32]. The specific form chosen here is…”

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

“…Several control strategies have been developed for purposeful manipulation of wave dynamics as the application of closed-loop or feedback-mediated control loops with and without delays [8][9][10][11] and open-loop control that includes external spatio-temporal forcing [10,[12][13][14], optimal control [15][16][17], and control by imposed geometric constraints and heterogeneities on the medium [18,19]. While feedback-mediated control relies on continuously monitoring of the system's state, open-loop control is based on a detailed knowledge of the system's dynamics and its parameters.…”