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Regular and chaotic bubble oscillations in periodically driven pressure fields. (English) Zbl 0636.76116

Summary: The motion of a single bubble in a periodically driven pressure field is examined from a geometric point of view using Poincaré maps. It is shown that the equations of motion can be transformed to a perturbation of a Hamiltonian system. The conditions determining nonlinear resonance are found; these correspond to subharmonic bifurcations. Further it is illustrated how the resonant response interacts with the nonresonant one to produce jump bifurcations. Results are also presented indicating that the periodic response undergoes a complex bifurcation sequence and a strange attractor forms. Finally it is demonstrated how the strange attractor disappears creating horseshoe maps that are associated with transient chaos. This gives some indication of the bifurcations that form the superstructure for single bubble oscillations.

MSC:

76T99 Multiphase and multicomponent flows
37N99 Applications of dynamical systems
76M99 Basic methods in fluid mechanics
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
35B32 Bifurcations in context of PDEs
70Sxx Classical field theories
26D15 Inequalities for sums, series and integrals
06B23 Complete lattices, completions
40A05 Convergence and divergence of series and sequences
40B05 Multiple sequences and series

Citations:

Zbl 0137.263
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References:

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