Nonlinear Dynamics

, Volume 94, Issue 4, pp 2373–2389 | Cite as

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

The exoskeleton, its internal structure and the missing fine substructure
  • Kálmán KlapcsikEmail author
  • Roxána Varga
  • Ferenc Hegedűs
Original Paper


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.


Keller–Miksis equation GPU programing AUTO boundary value problem solver Subharmonic topology Bi-parametric bifurcation structure 



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).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  • Kálmán Klapcsik
    • 1
    Email author
  • Roxána Varga
    • 1
  • Ferenc Hegedűs
    • 1
  1. 1.Department of Hydrodynamic Systems, Faculty of Mechanical EngineeringBudapest University of Technology and EconomicsBudapestHungary

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