Ovo web sjedište koristi tehnologiju "kolačića" (eng. cookie) da bi se korisnicima prikazao dinamičan i personaliziran sadržaj, te su oni nužni za ispravan rad. Ako nastavite pregledavati ove stranice, kolačići će biti korišteni u suradnji s vašim preglednikom Weba.
The notion of fractal dimension was born in the beginning of the 20th century in the works of Hemann Minkowski 1903, Felix Hausdorff 1919 and Georges Bouligand 1928. There are many other names for box dimension appearing in the literature, usually meaning the upper box dimension. One can encounter other equivalent names such as box counting dimension, Minkowski-Bouligand dimension, the Cantor-Minkowski order, Minkowski dimension, Bouligand dimension, Borel logarithmic rarefaction, Besicovitch-Taylor index, entropy dimension, Kolmogorov dimension, fractal dimension, capacity dimension, and limit capacity. The notion of Minkowski content is lesser known. Benoit Mandelbrot used it to define lacunarity as the reciprocal value of the content. A great impetus to oscillation theory was given by Claude Tricot with his research of fractal properties of smooth planar curves. In his monograph "Curves and Fractal Dimensions" he noticed that the box dimension can distinguish nonrectifiable smooth curves near the point of accumulation, while the Hausdorff dimension cannot.
Since 1970s the theory of fractal dimensions in dynamics has evolved into an independent branch of mathematics. Our research group has established the connection between the box dimension of a trajectory of dynamical systems and the bifurcation of the system.
Short description of the task performed by Croatian partner
The goals is to understand various aspects around the famous 16th Hilbert Problem (finding an upper bound for the number of limit cylces for a class of polynomial right-hand sides of a given order), which is the only unsolved problem among 23 of his problems from 1900 for the 20th century. The value of box dimension of the weak focus at the moment of Hopf-Takens bifurcation is directly related to the number of limit cycles that can be born, a fact established for the first time by D. Žubrinić and V. Županović in a joint paper published in Bulletin des sciences mathematiques in 2005. We expect that this principle can be established for other important bifurcation problems , like the saddle loop bifurcation, Hopf-Takens bifurcation at infinity, also in the case of discrete systems, etc.
The first goal is to analyse Poincare map near saddle loop of the planar vector field. That is, to relate the box dimension and the Minkowski content of an orbit of the discrete system generated by the Poincare map, and the number of limit cycles which could be produced by perturbation of the system.
Ovo web sjedište koristi tehnologiju "kolačića" (eng. cookie) da bi se korisnicima prikazao dinamičan i personaliziran sadržaj, te su oni nužni za ispravan rad. Ako nastavite pregledavati ove stranice, kolačići će biti korišteni u suradnji s vašim preglednikom Weba.
login
