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arXiv · 2610.28513

Resonant three-mode coupling in asteroseismology I: Bifurcations, chaos, and DAV white-dwarf outbursts

Zhao Guo·Gordon I. Ogilvie
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Abstract

Resonant three-mode coupling is a fundamental nonlinear process in stellar oscillations and other wave systems. We investigate the global dynamics of a linearly overstable parent mode coupled to two damped daughter modes. The principal control parameters are frequency detuning $Δ$ and daughter damping rate $Γ$, normalized by the parent growth rate. We combine high-resolution maps of the largest Lyapunov exponent in the $(Γ,Δ)$ plane with linear stability analysis, Floquet calculations, and numerical continuation. The resulting parameter space includes stable equilibria, simple and period-doubled limit cycles, chaotic attractors, formally unbounded solutions, and higher-period `shrimp' islands within chaotic regions. Successive period doublings lead to chaos, while boundaries of isolated periodic windows exhibit type-I intermittency. At large $Γ$, the main bifurcation boundaries approach straight lines, indicating that the dynamics is governed primarily by $r=Δ/(2Γ)$. We connect finite-$Γ$ bifurcations to an earlier one-dimensional return map. As $ε\rightarrow0$, this map recovers the principal period-doubling sequence, the onset of chaos near $r\simeq0.38$, and the asymptotic positions of prominent periodic windows. We also examine the effects of unequal daughter damping rates. Applying these results to sporadic outbursts in cool DAV white dwarfs, we find that estimated $g$-mode growth rates, damping rates, and detunings place resonant triplets in both periodic and chaotic regimes. Thus, instability of the three-mode equilibrium need not produce a regular limit cycle: deterministic chaos and intermittency may contribute to irregular outburst amplitudes and recurrence times.

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Zhao Guo, Gordon I. Ogilvie. 2026-10-20. Resonant three-mode coupling in asteroseismology I: Bifurcations, chaos, and DAV white-dwarf outbursts. https://arxiv.org/abs/2610.28513

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