Optimized Periodic Control of Chaotic Systems

In this work, we demonstrate the open-loop control of chaotic systems by means of optimized periodic signals. The use of such signals enables us to reduce control power significantly in comparison to simple harmonic perturbations. It is found that the stabilized periodic dynamics can be changed by small, specific alterations of the control signal. Thus, low power switching between different periodic states can be achieved without feedback. The robustness of the proposed control method against noise is discussed.Comment: 12 pages, uuencoded gzip-compressed postscript fil

High Dimensional Parameter Fitting of the Keller–Miksis Equation on an Experimentally Observed Dual-Frequency Driven Acoustic Bubble

A parameter identification technique of an underlying bubble model of an experimentally observed single bubble in a cluster under dual-frequency external forcing is presented. The measurements are carried out via high-speed camera recordings at a rate of 162750 frames per second. The used frequencies during the experiment are 25 kHz and 50 kHz. With a digital image processing technique, the measured bubble radius as a function of time is determined. The employed governing equation for the parameter fitting is the Keller–Miksis equation being a second order ordinary differential equation. The unknown four-dimensional parameter space is composed by the two pressure amplitudes, the phase shift of the dual-frequency driving and the equilibrium size of the bubble. In order to obtain an optimal parameter set within reasonable time, an in-house initial value problem solver is used running on a graphics processing unit (GPU). The error function measuring the distance between the numerical simulations and the measurement is based on the identification of the maximum bubble radii during each subsequent period of the external forcing. The results show a consistent estimation of both pressure amplitudes. The optima of phase shift and equilibrium bubble size are less significant due to a valley-like shape of the error function. Nevertheless, reasonable values are found that lead to estimations of pressure and temperature peaks during bubble collapse

Frequency Locking for Combustion Synthesis in Periodic Medium

Solutions of a 1-D free-interface problem modeling solid combustion front propagating in combustible mixture with periodically varying concentration of reactant exhibit classical phenomenon of mode locking. Numerical simulation shows a variety of locked periodic, quasi-periodic and chaotic solutions.Comment: 11 pages, 6 figure

Nonlinear Dynamics of a Single Ferrofluid-Peak in an Oscillating Magnetic Field

If a magnetic field normal to the surface of a magnetic fluid is increased beyond a critical value a spontaneous deformation of the surface arises (normal field instability). The instability is subcritical and leads to peaks of a characteristic shape. We investigate the neighborhood of this instability experimentally under the influence of a temporal modulation of the magnetic field. We use a small vessel, where only one peak arises. The modulation can either be stabilizing or destabilizing, depending on the frequency and amplitude. We observe a cascade of odd-numbered response-periods up to period 11, and also a domain of even-numbered periods. We propose a minimal model involving a cutoff-condition which captures the essence of the experimental observations. PACS: 47.20.-k, 47.20.Ky, 75.50.Mm Keywords: magnetic fluid; nonlinear oscillator; subharmonic response; surface instability;Comment: 13 pages, 12 Postscript figures, LaTeX, uses elsart.sty, to be published in Physica

GPU accelerated investigation of a dual-frequency driven nonlinear oscillator

The bifurcation structure of a dual-frequency driven, second order nonlinear oscillator (Keller–Miksis equation) is investigated by exploiting the high computational resources of professional GPUs. The numerical scheme of the applied initial value problem solver was the explicit, adaptive Runge–Kutta–Cash–Karp method with embedded error estimation using solutions of order 4 and 5. The four dimensional parameter space (amplitudes and frequencies of the driving) is explored by means of several high resolution bi-parametric plots with the amplitudes as control parameters at fixed frequencies. The resolution of the control parameter plane is 500 × 500 with 10 initial conditions at each parameter pair (altogether 2.5 million initial value problems in each bi-parametric plot). The program code for one fine parameter scan runs approximately 50 times faster on a Tesla K20 GPU (Kepler architecture) than on an Intel i7-4790 4 core CPU even applying double precision floating point operations

Fast jets from bubbles close to solid objects: examples from pillars in water to infinite planes in different liquids

The dynamics of a single, laser-induced cavitation bubble on top of a solid cylinder and right at a plane solid boundary is studiedboth experimentally and numerically. The most intriguing phenomenon that occurs for a millimeter sized bubble right at a flatsolid boundary in water is the formation of a fast jet that is directed towards the solid with a speed of the order of 1000 m/s.Paradoxically, in this setting, fast jet formation causally is related to the viscosity of the liquid.Thus, results from numericalsimulations with varying liquid viscosity and bubble size are presented. Bubble dynamics and jet formation mechanisms arediscussed. It is shown, that fast jet formation persists for a wide range of liquid viscosities, including e.g. 50 cSt silicone oil. Forbubbles generated close to the flat top of a long, thin cylinder the parameter space of initial distance to the cylinder, bubble size andcylinder radius is scanned numerically and partly compared to experiments. When the maximum radius of the bubble exceeds theone of the cylinder the bubble collapses in the form of a mushroom or can resemble a trophy, depending on the values of thegeometry parameters. Complex patterns of jet formation with jet speeds ranging from the order of a few hundred m/s to severalthousand m/s are found.The results represent a contribution to understand the behavior of bubbles collapsing close to solid surfaces,in particular, how thin, fast jets are generated