A. Hagberg et al., ORDER-PARAMETER EQUATIONS FOR FRONT TRANSITIONS - PLANAR AND CIRCULARFRONTS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 55(4), 1997, pp. 4450-4457
Near a parity breaking front bifurcation, small perturbations may reve
rse the propagation direction of fronts. Often this results in nonstea
dy asymptotic-motion, such as breathing and domain breakup. Exploiting
the time scale differences of an activator-inhibitor model and the pr
oximity to the front bifurcation, we derive equations of motion for pl
anar and circular fronts. The equations involve a translational degree
of freedom and an order parameter describing transitions between left
and right propagating fronts. Perturbations, such as a space dependen
t advective field or uniform curvature (axisymmetric spots), couple th
ese two degrees of freedom. In both cases this leads to a transition f
rom stationary to oscillating fronts as the parity breaking bifurcatio
n is approached. For axisymmetric spots, two additional dynamic behavi
ors are found: rebound and collapse.