Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance

Abstract

We consider the 2:1 internal resonances (such that íljX) and £l 2 -2£l l are natural frequencies) that appear in a nearly inviscid, axisymmetric capillary bridge when the slenderness A is such that 0< A<77 (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 £l l and Cl 2 , an d 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 Cl 2 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 A -2.23 and involves the fifth and eighth eigenmodes. If, instead, the eigenmode associated with Cl 2 is °dd, and only one of the eigenmodes associated with ílj and Cl 2 is directly excited, then quadratic terms are absent in the normal form and the associated dynamics is seen to be fairly simple. O 1998 American Institute ofPhysics. [S1070-6631(98)

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