arXiv subjects

Vesna Županović

Publications and source records attributed to Vesna Županović.

6 recordsLinked to original sources

arXiv2025

Fractal analysis of slow-fast and regular systems: A survey of recent results and future perspectives

We survey recent developments in fractal analysis of regular and slow-fast dynamical systems using Minkowski dimension. Our focus is on spiral trajectories near monodromic limit periodic sets in regular systems and entry-exit sequences in slow-fast systems with degenerate singularities. For regular systems, we recall connections between Minkowski dimension and cyclicity. Key results include fractal classifications of weak foci, degenerate foci, and polycycles, where dimensional relationships predict limit cycle birth. For slow-fast systems, we survey the coordinate-free fractal methodology analyzing slow-fast Hopf points and canard cycles through slow divergence integrals and entry-exit sequences. The Minkowski dimension takes discrete values yielding upper bounds for the number of limit cycles without normal form transformations. The fractal approach provides computational advantages, works with original coordinates, and reveals geometric structures underlying bifurcation phenomena. Applications span neuroscience, chemistry, population dynamics, and climate modeling. We also discuss extensions to piecewise smooth and three-dimensional systems.

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arXiv2024

Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits

We consider generic 1-parameter unfoldings of parabolic vector fields. It is known that the box dimension of orbits of their time-one maps is discontinuous at the bifurcation value. Here, we expand asymptotically the Lebesgue measure of the epsilon-neighborhoods of orbits of the time-one maps in a Chebyshev scale, uniformly with respect to the bifurcation parameter. We use the so-called Ecalle-Roussarie-type compensators. We read from the expansion the number of hyperbolic points born in the unfolding of the parabolic point (i.e. the codimension of the bifurcation). We consider generic analytic 1-parameter unfoldings of saddle-node germs of analytic vector fields on the real line, their time-one maps and the Lebesgue measure of $\varepsilon$-neighborhoods of the orbits of these time-one maps.The box dimension of an orbit gives the asymptotics of the principal term of this Lebesgue measure and it is known that it is discontinuous at bifurcation parameters. In order to recover continuous dependence of the asymptotics on the parameter, here we expand asymptotically the Lebesgue measure of $\varepsilon$-neighborhoods of orbits of time-one maps in a Chebyshev system, uniformly with respect to the bifurcation parameter. We use Écalle-Roussarie-type compensators. We show how the number of fixed points of the time-one map born in the universal analytic unfolding of the parabolic point corresponds to the number of terms vanishing in this uniform expansion of the Lebesgue measure of $\varepsilon$-neighborhoods of orbits.

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arXiv2021

The box dimension of degenerate spiral trajectories of a class of ordinary differential equations

In this paper we initiate the study of the box dimension of degenerate spiral trajectories of a class of ordinary differential equations. A class of singularities of focus type with two zero eigenvalues (nilpotent or more degenerate) has been studied. We find the box dimension of a polynomial degenerate focus of type $(n,n)$ by exploiting the well-known fractal results for $α$-power spirals. In the general $(m,n)$ case, we formulate a conjecture about the box dimension of a degenerate focus. Further, we reduce the fractal analysis of planar nilpotent contact points to the study of the box dimension of a slow-fast spiral generated by their "entry-exit" function. There exists a bijective correspondence between the box dimension of the slow-fast spiral and the codimension of contact points. We also construct a three-dimensional vector field that contains a degenerate spiral, called an elliptical power spiral, as a trajectory.

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arXiv2016

Characteristic box dimension of unit-time map near nilpotent singularity of planar vector field and applications

This article shows how box dimension of the unit-time map can be used in studying the multiplicity of nilpotent singularities of planar vector fields. Using unit-time map on the characteristic curve of nilpotent singularity we define characteristic map and characteristic box dimension of the unit-time map. We study connection between the box dimension of discrete orbits generated by the unit-time map of planar vector fields on the characteristic or invariant curves, and the multiplicity of singularities. Nilpotent singularities which are studied are nilpotent node, focus and cusp. Also, we show how box dimension of the Poincar\' e map near the nilpotent focus on the characteristic curve reveals the upper bound for cyclicity. Moreover, we study the characteristic box dimension of nilpotent cusp at infinity which is connected to the order of cusp. At the end, we applied the results of box dimension of unit-time map to the Bogdanov-Takens bifurcation.

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arXiv2015

Fractal analysis of Hopf bifurcation at infinity

Using geometric inversion with respect to the origin we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the Riemann sphere. We study its basic properties, and apply it to the study of the Hopf-Takens bifurcation at infinity.

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arXiv2009

Box dimension of trajectories of some discrete dynamical systems

We study the asymptotics, box dimension, and Minkowski content of trajectories of some discrete dynamical systems. We show that under very general conditions, trajectories corresponding to parameters where saddle-node bifurcation appears have box dimension equal to 1/2, while those corresponding to period-doubling bifurcation parameter have box dimension equal to 2/3. Furthermore, all these trajectories are Minkowski nondegenerate. The results are illustrated in the case of logistic map.

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