Bistable fronts in periodic media: phase reduction and realization of circle flows
Abstract
We establish a phase reduction and a converse realization theorem for front dynamics in one-dimensional periodic reaction-diffusion-advection equations. For sufficiently large periods under local bistability, and sufficiently small periods when the homogenized equation admits a stationary front, we construct strictly ordered invariant front families that are normally exponentially attracting and locally complete for entire orbits. Modulo spatial-period translations, the dynamics on each family reduces exactly to a circle flow: rotations correspond to propagating pulsating fronts, equilibria to stationary fronts, and connecting orbits to transition fronts between stationary fronts. Conversely, every smooth circle flow can be realized, up to time-preserving topological conjugacy, as the quotient dynamics of such a family for a periodic, locally bistable equation with constant diffusion and no advection. Realizations with a Cantor set of stationary phases and a connecting orbit in each complementary interval illustrate the richness of front dynamics in these media. We also obtain first-order phase-velocity expansions in both period limits, quantitative propagation laws near zero frozen speed, and global exponential orbital stability of propagating pulsating fronts.
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BibTeX
@misc{ding2026bistable,
title = {{Bistable fronts in periodic media: phase reduction and realization of circle flows}},
author = {Weiwei Ding and Xing Liang and Linfeng Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34228},
eprint = {2610.34228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34228},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bistable fronts in periodic media: phase reduction and realization of circle flows
AU - Weiwei Ding
AU - Xing Liang
AU - Linfeng Xu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34228
UR - https://arxiv.org/abs/2610.34228
AB - We establish a phase reduction and a converse realization theorem for front dynamics in one-dimensional periodic reaction-diffusion-advection equations. For sufficiently large periods under local bistability, and sufficiently small periods when the homogenized equation admits a stationary front, we construct strictly ordered invariant front families that are normally exponentially attracting and locally complete for entire orbits. Modulo spatial-period translations, the dynamics on each family reduces exactly to a circle flow: rotations correspond to propagating pulsating fronts, equilibria to stationary fronts, and connecting orbits to transition fronts between stationary fronts. Conversely, every smooth circle flow can be realized, up to time-preserving topological conjugacy, as the quotient dynamics of such a family for a periodic, locally bistable equation with constant diffusion and no advection. Realizations with a Cantor set of stationary phases and a connecting orbit in each complementary interval illustrate the richness of front dynamics in these media. We also obtain first-order phase-velocity expansions in both period limits, quantitative propagation laws near zero frozen speed, and global exponential orbital stability of propagating pulsating fronts.
ER -