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Bi-parametric topology of subharmonics of an asymmetric bubble oscillator at high dissipation rate

The exoskeleton, its internal structure and the missing fine substructure

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Abstract

The subharmonic topology of a nonlinear, asymmetric bubble oscillator (Keller–Miksis equation) in glycerine is investigated in the parameter space of its external excitation (frequency and pressure amplitude). The bi-parametric investigation revealed that the exoskeleton of the topology can be described as the composition of U-shaped subharmonics of periodic orbits. The fine substructure (higher-order ultra-subharmonic resonances) usually appearing via the well-known period n-tupling phenomenon is completely missing due to the high dissipation rate of the viscous liquid. Moreover, a complex internal structure of the subharmonics has been found, which are composed by interconnected bifurcation blocks (in a zig-zag pattern) each describing the skeleton of a shrimp-shaped domain. The employed numerical techniques are the combination of an in-house initial value problem solver written in C++/CUDA C to harness the high processing power of professional graphics cards, and the boundary value problem solver AUTO to compute periodic orbits directly regardless of their stability.

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Acknowledgements

This paper was supported by the ÚNKP-17-3-I New National Excellence Program of the Ministry of Human Capacities, by the János Bolyai Research Scholarship of Hungarian Academy of Sciences and by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Water science & Disaster Prevention research area of Budapest University of Technology and Economics (BME FIKP-VÍZ).

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  1. Department of Hydrodynamic Systems, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, P.O. Box 91, Budapest, 1521, Hungary

    Kálmán Klapcsik, Roxána Varga & Ferenc Hegedűs

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  1. Kálmán Klapcsik
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  3. Ferenc Hegedűs
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Klapcsik, K., Varga, R. & Hegedűs, F. Bi-parametric topology of subharmonics of an asymmetric bubble oscillator at high dissipation rate. Nonlinear Dyn 94, 2373–2389 (2018). https://doi.org/10.1007/s11071-018-4497-2

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