Naslov
Fractal analysis of bifurcations of dynamical systems

Autori
Vesna Županović

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Theoretical and computational methods in dynamical systems and fractal geometry / - Maribor : Univerza v Mariboru, 2015, 40-42

Skup
Theoretical and computational methods in dynamical systems and fractal geometry

Mjesto i datum
Maribor, Slovenija, 07.04.2015. - 11.04.2015

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
dyamical system; fractal geometry; bifurcation; oscillatory integral

Sažetak
In this talk I would like to give an overview of results concerning fractal analysis of dynamical systems, obtained by scienti c group at University of Zagreb and our collaborators. Bifurcations of limit cycles are related to the 16th Hilbert problem. It asks for an upper bound or the number of limit cycles, of polynomial vector elds in the plane, as a function of the degree of the vector eld. The problem is still open. It is of special interest to determine how many limit cycles can bifurcate from a given limit periodic set in a generic unfolding. This number is called the cyclicity of the limit periodic set. The cyclicity is classically obtained by studying the multiplicity of xed points of Poincare map. We establish a relation between cyclicity of a limit periodic set of a planar system and fractal properties of the Poincare map of a trajectory of the system. A natural idea is that higher density of orbits reveals higher cyclicity. The study of density of orbits is where fractal analysis is applied. Classical fractal analysis associates box dimension and Minkowski content to bounded sets. They measure the density of accumulation of a set, see [10]. In the paper [11], the cyclicity of weak foci and limit cycles is directly related to the box dimension of any trajectory. It was discovered that the box dimension of a spiral trajectory of weak focus signals a moment of Hopf and Hopf- Takens bifurcation. The result was obtained using Takens normal form. In [12], box dimension of spiral trajectories of weak focus was related to the box dimension of its Poincare maps. Results were based on [2] and [3]. This article also shows that generic bifurcations of 1-dimensional discrete systems are characterised by the box dimension of orbits. Fractal analysis of Hopf bifurcation for discrete dynamical systems, called Neimark-Sacker bifurcation, has been completed in [4]. In the above continuous cases, the Poincare map was di erentiable, which 1 was crucial for relating the box dimension and the cyclicity of a limit periodic set. The problem in hyperbolic polycycle case is that the Poincare map in not di erentiable, but the family of maps in generic bifurcations has an asymptotic development in a so-called Chebyshev scale. We introduced in [5] the appropriate generalizations of box dimension, depending on a particular scale for a given problem. The cyclicity was concluded using the generalized box dimension in the case of saddle loops. The box dimension has been read from the leading term of asymptotic expansion of area of "- neighborhoods of orbits. If we go further into the asymptotic expansion we can make formal classi cation of parabolic di eo- morphisms using fractal data given in the expansion, see [7], and also [8]. Analogously it is possible to study singularities of maps, see [1]. We study geometrical representation of oscillatory integrals with analytic phase function and smooth amplitude with compact support. Geometrical and fractal properties of the curves de ned by oscillatory integral depend on type of critical point of the phase. Methods in [9] include Newton diagrams and resolution of singularities.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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