Efficient Rigorous Continuation via Chebyshev Series Expansion I
Abstract
We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted ℓ1 space, giving a finer control than the typical C0-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn--Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada--Kawasaki--Teramoto system.
Related
- An efficient and rigorous criterion for the admissible time step of deforming grids in generalized Lagrangian and ALE methods
- Uncertainty Quantification for the Fission Matrix Method: A Rigorous Mathematical Framework and Computationally Efficient Alternatives
- Point-to-ellipse series -- Part II Expansion about the center
- A new six parameters generalized Weibull-Kumaraswamy-{Lomax) Distribution with a Novel Linear Series Expansion of Its Probability Density
- Efficient Solution of Third-Order ODEs with Diagonally Implicit RK Method Using Tri-Colored Tree and B-series Analysis
BibTeX
@misc{breden2026efficient,
title = {{Efficient Rigorous Continuation via Chebyshev Series Expansion I}},
author = {Maxime Breden and Olivier Hénot},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22689},
eprint = {2609.22689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22689},
}
RIS
TY - UNPB
TI - Efficient Rigorous Continuation via Chebyshev Series Expansion I
AU - Maxime Breden
AU - Olivier Hénot
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22689
UR - https://arxiv.org/abs/2609.22689
AB - We study the global continuation of solution manifolds arising in dynamical systems. We present a rigorous continuation method based on a Chebyshev series expansion of the solution manifold. The branch is first approximated by a high-order Chebyshev interpolation polynomial, and an explicit error bound is then obtained by verifying the contraction of a quasi-Newton operator near this approximation. The contraction is formulated on a weighted ℓ^1 space, giving a finer control than the typical C^0-error bound obtained from the uniform contraction theorem. In fact, the latter follows directly from our contraction operator. Furthermore, we discuss how our strategy applies naturally to pseudo-arclength continuation, where the continuation parameter fails to provide a valid local coordinate, and extends to multi-parameter continuation. Lastly, we detail two applications in which we compute a two-parameter family of steady-states for the Cahn--Hilliard equation, and a one-parameter family of steady-states undergoing saddle-node bifurcations for the Shigesada--Kawasaki--Teramoto system.
ER -