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Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance
Mancebo, Francisco J. and Nicolás, José A. and Vega de Prada, José Manuel
Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance.
"Physics of Fluids", v. 10
We consider the 2:1 internal resonances (such that Ω1>0 and Ω2 ≃ 2Ω1 are natural frequencies) that appear in a nearly inviscid, axisymmetric capillary bridge when the slenderness Λ is such that 0<Λ<π (to avoid the Rayleigh instability) and only the first eight capillary modes are considered. A normal form is derived that gives the slow evolution (in the viscous time scale) of the complex amplitudes of the eigenmodes associated with Ω1 and Ω2, and consists of two complex ODEs that are balances of terms accounting for inertia, damping, detuning from resonance, quadratic nonlinearity, and forcing. In order to obtain quantitatively good results, a two-term approximation is used for the damping rate. The coefficients of quadratic terms are seen to be nonzero if and only if the eigenmode associated with Ω2 is even. In that case the quadratic normal form possesses steady states (which correspond to mono- or bichromatic oscillations of the liquid bridge) and more complex periodic or chaotic attractors (corresponding to periodically or chaotically modulated oscillations). For illustration, several bifurcation diagrams are analyzed in some detail for an internal resonance that appears at Λ ≃ 2.23 and involves the fifth and eighth eigenmodes. If, instead, the eigenmode associated with Ω2 is odd, and only one of the eigenmodes associated with Ω1 and Ω2 is directly excited, then quadratic terms are absent in the normal form and the associated dynamics is seen to be fairly simple.
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