三次判别式作为Duffing动力学的组织原理
The cubic discriminant as an organizing principle for Duffing dynamics
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中文总结 AI 辅助
本文提出以三次判别式为组织原理，统一解析周期受迫Duffing振子的动力学，给出阱间/阱内阈值并揭示无质量极限作为有限质量全局分岔的中心。
中文摘要 AI 辅助
我们提出了周期性受迫Duffing振子动力学的一个解析组织原理。在无质量极限下，该系统简化为一个含时三次方程，其判别式和雅可比行列式对典型Duffing区域进行分类，并确定瞬时平衡点的数目和稳定性。对于双阱振子，相同的构造给出了一个精确的强迫阈值$F_{\mathrm{bif}}$，用于区分阱内和阱间动力学。我们证明，这个代数量支配着远超出其起源奇异极限的有限质量动力学：阱内与阱间运动之间的边界在$m\to0$时收敛于$F_{\mathrm{bif}}$，并且导致不对称阱间态的分岔点分岔族在公共点处累积。数值模拟进一步展示了退化模型的不连续跳跃如何在有限质量下被正则化为短振荡瞬态，且这些瞬态在$m\to0$时收缩。这些结果揭示了无质量Duffing方程作为完整有限质量动力学的组织中心，并建立了退化问题的代数结构与振子全局分岔结构之间的直接联系。
英文摘要
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 $F_{\mathrm{bif}}$ 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 $F_{\mathrm{bif}}$ as $m\to0$, 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 $m\to0$. 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.
发表机构
- Lodz University of Technology（罗兹理工大学）
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