Rigorous verification of period-doubling bifurcations in maps
Abstract
We present a computer-assisted approach for proving period-doubling bifurcations in one-parameter families of maps x ↦ f(x,λ) on ℝn. We solve a square augmented system, called the period-doubling map, whose unknowns are the fixed point, the parameter, the right eigenvector attached to the eigenvalue −1, the left eigenvector with its eigenvalue, and the vector which describes the center manifold to second order. A contraction argument gives a solution close to a numerical approximation, at which the Jacobian is invertible. The determinant of this Jacobian factors into a simple expression, so that it is nonzero exactly when $1$ is not an eigenvalue and −1 is a simple eigenvalue crossing the unit circle transversally. Apart from the sign of the cubic normal-form coefficient, computed afterwards, it remains to check with Gershgorin disks that no other eigenvalue lies on the unit circle. The method is applied to a $10$-mode Galerkin projection of the Kot--Schaffer growth-dispersal model, for which we prove a supercritical period-doubling bifurcation.
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BibTeX
@misc{henot2026rigorous,
title = {{Rigorous verification of period-doubling bifurcations in maps}},
author = {Olivier Hénot and Jean-Philippe Lessard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22986},
eprint = {2610.22986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22986},
}
RIS
TY - UNPB
TI - Rigorous verification of period-doubling bifurcations in maps
AU - Olivier Hénot
AU - Jean-Philippe Lessard
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22986
UR - https://arxiv.org/abs/2610.22986
AB - We present a computer-assisted approach for proving period-doubling bifurcations in one-parameter families of maps x ↦ f(x,λ) on ℝ^n. We solve a square augmented system, called the period-doubling map, whose unknowns are the fixed point, the parameter, the right eigenvector attached to the eigenvalue −1, the left eigenvector with its eigenvalue, and the vector which describes the center manifold to second order. A contraction argument gives a solution close to a numerical approximation, at which the Jacobian is invertible. The determinant of this Jacobian factors into a simple expression, so that it is nonzero exactly when $1$ is not an eigenvalue and −1 is a simple eigenvalue crossing the unit circle transversally. Apart from the sign of the cubic normal-form coefficient, computed afterwards, it remains to check with Gershgorin disks that no other eigenvalue lies on the unit circle. The method is applied to a $10$-mode Galerkin projection of the Kot--Schaffer growth-dispersal model, for which we prove a supercritical period-doubling bifurcation.
ER -