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The effect of high viscosity on the evolution of the bifurcation set of a periodically excited gas bubble

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  • Klapcsik, Kálmán
  • Hegedűs, Ferenc

Abstract

In this study, a nonlinear investigation of a periodically driven gas bubble in glycerine is presented. The bifurcation structure of the bubble oscillator (Keller–Miksis equation) is explored in the pressure amplitude-frequency parameter plane of the excitation by means of initial (high resolution bi-parametric plots) and boundary value problem solvers at various ambient temperatures. The range of the applied temperature covers two orders of magnitude difference in the liquid viscosity which is the main damping factor of the system. Therefore, the evolution of the harmonic and ultraharmonic resonances are presented starting with an overdamped behaviour (there are no resonances in the parameter space) and ending up with a fully developed bifurcation superstructure. The results reveal a complex period bubbling mechanism organized in a Farey-tree; inside each bubble a fine substructure of alternating chaotic and periodic bands exist. The description of the bifurcation structure presented throughout the paper can help to understand the mechanism of dissipation on the behaviour of nonlinear systems in more detail.

Suggested Citation

  • Klapcsik, Kálmán & Hegedűs, Ferenc, 2017. "The effect of high viscosity on the evolution of the bifurcation set of a periodically excited gas bubble," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 198-208.
    • Handle: RePEc:eee:chsofr:v:104:y:2017:i:c:p:198-208
    DOI: 10.1016/j.chaos.2017.08.022
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    References listed on IDEAS

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    1. Varga, Roxána & Paál, György, 2015. "Numerical investigation of the strength of collapse of a harmonically excited bubble," Chaos, Solitons & Fractals, Elsevier, vol. 76(C), pages 56-71.
    2. Cabeza, Cecilia & Briozzo, Carlos A. & Garcia, Rodrigo & Freire, Joana G. & Marti, Arturo C. & Gallas, Jason A.C., 2013. "Periodicity hubs and wide spirals in a two-component autonomous electronic circuit," Chaos, Solitons & Fractals, Elsevier, vol. 52(C), pages 59-65.
    3. Behnia, Sohrab & Jafari, Amin & Soltanpoor, Wiria & Jahanbakhsh, Okhtay, 2009. "Nonlinear transitions of a spherical cavitation bubble," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 818-828.
    4. Medeiros, E.S. & de Souza, S.L.T. & Medrano-T, R.O. & Caldas, I.L., 2011. "Replicate periodic windows in the parameter space of driven oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 982-989.
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