Hegedűs, Ferenc

Lauterborn, Werner

Mettin, Robert

Parlitz, Ulrich

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ENOC 2017, June 25-30, 2017, Budapest, HungaryInteraction of period-1 orbits in a dual-frequency driven asymmetric nonlinearoscillatorFerenc Hegedu˝s∗, Werner Lauterborn∗∗, Ulrich Parlitz ∗∗∗ and Robert Mettin∗∗∗Department of Hydrodynamic Systems, Budapest University of Technology and Economics,Budapest, Hungary∗∗Drittes Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany∗∗∗Biomedical Physics Group, Max Planck Institute for Dynamics and Self-Organization, Göttingen,Germany and Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Göttingen,GermanySummary. The bifurcation structure of period-1 solutions of a dual-frequency driven asymmetric nonlinear oscillator (Keller–Miksisequation, describing bubble dynamics) is examined. The applied frequencies are ω1/ω0 = 2 and ω2/ω0 = 3, where ω0 is the linear,undamped resonance frequency of the system. The control parameters were the amplitudes of the driving. Due to the specific choice ofthe frequency ratio and Poincaré section, the period-2 and period-3 orbits (corresponding to the monofrequency drivings of ω1/ω0 = 2and ω2/ω0 = 3, respectively) are decomposed into a multitude of period-1 orbits. The combination of the two frequency componentsresults in a complex bifurcation structure and interaction of these solutions in the parameter plane of the driving amplitudes.IntroductionDual-frequency driven nonlinear oscillators are thoroughly investigated during the last decades. From the early 80’s,researchers paid special attention to quasiperiodically driven systems (using incommensurable frequencies) producing aspecial kind of solution called strange nonchaotic attractor [1]. It is strange due to its fractal nature but nonchaotic becauseit does not possess positive Lyapunov exponents. The application of a second frequency component is also successfullyused for chaos control to eliminate unpredictable behaviour of certain systems [2]. Moreover, vibrational resonanceinduced by the application of two distant frequencies plays an important role in many experiments [3].The bifurcation structure in the two-dimensional parameter space of the driving with fixed, commensurable frequencies,however, is less elaborated. This is the main topic of the present study. The employed Keller–Miksis equation is a secondorder ordinary nonlinear differential equation widely used to model the oscillation of a single spherical gas bubble in aliquid domain irradiated with high intensity ultrasound [4]. This model was successfully used to improve the efficiencyof sonochemistry, material chemistry and food processing [5]. This study is to be understood as a very first research stepto possible further improvement of such applications.Mathematical modelThe mathematical model describing the radial oscillation of a single spherical bubble (Keller–Miksis equation) can bewritten as (1−R˙cL)RR¨+(1−R˙3cL)32R˙2 =(1 +R˙cL+RcLddt)(pL − P∞ − p∞(t))ρL, (1)pL = pG + pV −2σR− 4µLR˙R, pG =(P∞ − pV +2σRE)(RER)3n, (2)where R(t) is the time dependent bubble radius. For the details of the model see [4]. The parameter values during thecomputations (gas bubble in water) were as follows: liquid density ρL = 997.1 kg/m3, sound speed cL = 1497.3m/sand viscosity µL = 8.902−4Pa s; vapour pressure pV = 3166.8Pa; surface tension σ = 0.072N/m; ambient (static)pressure P∞ = 1bar; bubble size RE = 10µm; polytropic exponent n = 1.4 (adiabatic behaviour). The dual-frequencydriving of the systemp∞(t) = pA1 sin(ω1t) + pA2 sin(ω2t) (3)is a time varying pressure field, where pA1 and pA2 are the pressure amplitudes (also control parameters); ω1 = 3ω0 andω2 = 2ω0 are the corresponding angular frequencies. Here ω0 is the linear undamped resonance frequency [4].By introducing dimensionless time τ = t ω1/2pi and bubble radius y1 = R/RE , the dimensionless bubble wall velocityy2 and the dual-frequency driving become y2 = R˙ 2pi/(REω1) andp∞(τ) = pA1 sin(2piτ) + pA2 sin(2piω2/ω1τ), (4)respectively. Observe that the periods of the first and second components are T1 = 1 and T2 = ω1/ω2 = 1.5, respectively.The periodicity of the dual frequency driving is T = 3 (the smallest integer multiple of both T1 and T2), which is used fora global Poincaré map. Therefore, in the specific case of pA1 = 0 only points after every second real driving period aresampled, while in case of pA2 = 0 only after every third real driving period.ENOC 2017, June 25-30, 2017, Budapest, HungaryBifurcation structure of period-1 orbitsTo obtain a global picture about the co-existing attractors in the pA1-pA2 bi-parametric plane, 10 initial value problems(IVP) were solved with random initial conditions at each parameter pair. The resolution of the pressure amplitudes were∆p = 0.01 bar in both directions. After the initial transient, a number of 128 Poincaré points were recorded. To solve the2.5 million IVPs within reasonable time, an in-house numerical code was used to exploit the huge computational capacityof GPUs (Nvidia Tesla K20m), with which the computational time took only 25.7 h. The integration algorithm was theadaptive Runge–Kutta–Cash–Karp method.The left hand side of Fig. 1 is a condensed view of the found attractors, where the colorbar represents periodicities up toperiod-9 (the black region is a mixture of chaotic and higher periodic solutions). The overlapping periodic domains showrich dynamics, from which only the period-1 solutions (grey area) are investigated in more details. The middle of Fig. 1shows the number of the co-existing period-1 solutions, where the overlapping domains indicate a complex structure ofperiod-1 attractors hidden in the grey area in Fig. 1 left which indicates only the presence of at least 1 period-1 solution.Figure 1: Left: Periodicities of the co-existing attractors. Middle: Number of the co-existing period-1 solutions. Right: Three-dimensional representation of the second component of the Poincaré section points P (y2) over the pA1-pA2 plane.The period-1 structure is more visible in the 3D representation of the second component of the Poincaré section pointsP (y2) (dimensionless bubble wall velocity) over the pA1-pA2 plane (Fig. 1 right) where the period-1 orbits form surfaces.The solutions composing the black surface PE1originates from the equilibrium solution denoted by E (observe that thevelocity y2 = 0 at pA1,2 = 0). If pA1 = 0, an originally period-2 solution is decomposed into two period-1 solutionsP 12and P 22emerged via a pitchfork bifurcation (PF), since T = 2T2 is used for a global Poincaré section instead of T2itself (see the discussion in the previous section). Similarly, in case of pA2 = 0, an originally period-3 solution, appearedthrough a saddle-node (SN) bifurcation, splits into three period-1 solutions P 13 , P 23 and P 33 (T = 3T1). The layers ofperiod-1 orbits marked by PE1and P 33are "standalone" surfaces. Observe, however, that there are smooth transformationsbetween following pairs of decomposed orbits: P 12 -P 13 and P 22 -P 23 .DiscussionIt has been shown that the period-1 orbits form multiple surfaces in the pA1-pA2 parameter plane. This is the consequenceof the specific choice of the global Poincaré section, which decomposes the period-2 and period-3 orbits into severalperiod-1 solutions under monofrequency driving. This is not surprising, and the difference between the decomposedsolutions is only a shift in time by the corresponding monofrequency driving period. Under dual-frequency excitation,however, these solutions become distinct. Moreover, an interaction between the originally period-2 and period-3 solutioncan be observed via the splitting mechanism, see again Fig. 1 right. Up to our knowledge, this behaviour in dual-frequencydriven system has not been observed previously.AcknowledgementThis paper was supported by the Deutsche Forschungsgemeinschaft (DFG) grant no. ME 1645/7-1, and by the JánosBolyai Research Scholarship of the Hungarian Academy of Sciences.References[1] Grebogi, C., Ott, E., Pelikan, S., Yorke, J. A. (1984) Strange attractors that are not chaotic. Physica D 13:261-268.[2] Behnia, S., Sojahrood, A. J., Soltanpoor, W., Jahanbakhsh, O. (2009) Suppressing chaotic oscillations of a spherical cavitation bubble throughapplying a periodic perturbation. Ultrason. Sonochem. 16:502-511.[3] Jeyakumari., S., Chinnathambi, V., Rajasekar, S., Sanjuan, M. A. F. (2011) Vibrational resonance in an asymmetric Duffing oscillator. Int. J.Bifurcat. Chaos 21:275-286.[4] Lauterborn, W. and Kurz, T. (2010) Physics of bubble oscillations. Rep. Prog. Phys. 73:106501.[5] Leonelli, C., Mason, T. J. (2010) Microwave and ultrasonic processing: Now a realistic option for industry. Chem. Eng. Process. 49:885-900.

Interaction of period-1 orbits in a dual-frequency driven asymmetric nonlinear oscillator

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