The cubic discriminant as an organizing principle for Duffing dynamics
Abstract
We present an analytical organizing principle underlying the dynamics of the periodically forced Duffing oscillator. In the massless limit the system reduces to a time-dependent cubic equation whose discriminant and Jacobian classify the canonical Duffing regimes and determine the number and stability of instantaneous equilibria. For the double-well oscillator, the same construction gives an exact forcing threshold Fbif separating intra-well and inter-well dynamics. We show that this algebraic quantity governs the finite-mass dynamics far beyond the singular limit from which it originates: the boundary between intra-well and inter-well motion converges to Fbif as m→0, and the families of pitchfork bifurcations responsible for asymmetric inter-well states accumulate at a common point. Numerical simulations further show how the discontinuous jumps of the degenerate model are regularized at finite mass into short oscillatory transients that shrink as m→0. These results reveal the massless Duffing equation as the organizing center of the full finite-mass dynamics and establish a direct link between the algebraic structure of the degenerate problem and the global bifurcation structure of the oscillator.
Versions
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BibTeX
@misc{jaros2026cubic,
title = {{The cubic discriminant as an organizing principle for Duffing dynamics}},
author = {Patrycja Jaros and Przemyslaw Perlikowski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26719},
eprint = {2609.26719},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26719},
}
RIS
TY - UNPB
TI - The cubic discriminant as an organizing principle for Duffing dynamics
AU - Patrycja Jaros
AU - Przemyslaw Perlikowski
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26719
UR - https://arxiv.org/abs/2609.26719
AB - We present an analytical organizing principle underlying the dynamics of the periodically forced Duffing oscillator. In the massless limit the system reduces to a time-dependent cubic equation whose discriminant and Jacobian classify the canonical Duffing regimes and determine the number and stability of instantaneous equilibria. For the double-well oscillator, the same construction gives an exact forcing threshold F_bif separating intra-well and inter-well dynamics. We show that this algebraic quantity governs the finite-mass dynamics far beyond the singular limit from which it originates: the boundary between intra-well and inter-well motion converges to F_bif as m→0, and the families of pitchfork bifurcations responsible for asymmetric inter-well states accumulate at a common point. Numerical simulations further show how the discontinuous jumps of the degenerate model are regularized at finite mass into short oscillatory transients that shrink as m→0. These results reveal the massless Duffing equation as the organizing center of the full finite-mass dynamics and establish a direct link between the algebraic structure of the degenerate problem and the global bifurcation structure of the oscillator.
ER -