Accuracy of the box-counting algorithm for numerical computation of the fractal exponents is investigated. To this end several sample mathematical fractal sets are analyzed. It is shown that the standard deviation obtained for the fit of the fractal scaling in the log-log plot strongly underestimates the actual error. The real computational error was found to have power scaling with respect to the number of data points in the sample (n_{tot}). For fractals embedded in two-dimensional space the error is larger than for those embedded in one-dimensional space. For fractal functions the error is even larger. Obtained formula can give more realistic estimates for the computed generalized fractal exponents' accuracy.
The considered chaotic oscillator consists of an amplifier, 2nd order LC resonator, Schottky diode and an extra capacitor in parallel to the diode. The diode plays the role of a nonlinear device. Chaotic oscillations are demonstrated numerically and experimentally at low as well as at high megahertz frequencies, up to 250 MHz.
Convection in horizontally vibrated granular systems is significant for scientists and engineers for their importance in the field of mining, geo-physics, and pharmaceutical etc. This research work studied three types of convection rolls, "Homogeneous convection roll", "lower-right diagonal convection roll" and "upper-right diagonal convection roll" which occurred in a square container filled with binary granular particles mixture of sized d=(4.0±0.2) mm and d=(8.0±0.2) mm. Container was vibrated horizontally with low frequencies f and low dimensionless acceleration Γ. Helical movement was observed along the walls perpendicular to direction of motion while straight-line movement along the walls horizontal to direction of motion. Helical and straight-line movements of particles along the walls are the part of convection rolls. A heap appeared due to vibration, which has dominant effect on the convection rolls. Heap position is function of frequency f and dimensionless accelerations Γ.
We analyze 2-}dimensional chaotic forms resulting from very simple systems based on two chaotic characteristics that is rotation and parallel movement or translation in geometric terms. Reflection is another alternative, along with rotation, for several interesting chaotic formations. Rotation and translation are very common types of movements in the world around us. Chaotic or non-chaotic forms arise from these two main generators. The rotation-translation chaotic case presented is based on the theory we analyzed in the book and in the paper. An overview of the chaotic flows in rotation-translation is given. There is observed the presence of chaos when discrete rotation-translation equation forms are introduced. In such cases the continuous equations analogue of the discrete cases is useful. Characteristic cases and illustrations of chaotic attractors and forms are analyzed and simulated. The analysis of chaotic forms and attractors of the models presented is given along with an exploration of the characteristic or equilibrium points. Applications in the fields of astronomy-astrophysics (galaxies), chaotic advection (the sink problem) and Von Karman streets are presented.
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