Vesna Županović

Fractal dimensions in dynamics

This is an invited article for the Encyclopedia of Mathematical Physics, published by Elsevier in Oxford in 2006. We describe some basic methods of fractal analysis in dynamics. A special emphasis is on the computation of Hausdorff and box dimensions. In particular, we present dimension results for the logistic map, Smale horseshoe, Lorenz and Henon attractors, Julia and Mandelbrot sets, spiral trajectories, attractors appearing in infinite-dimensional systems, and Brownian motion trajectories.

Generalized Fresnel integrals and fractal properties of related spirals

Abstract We obtain a new asymptotic expansion of generalized Fresnel integrals x ( t ) = ∫ 0 t cos q ( s ) d s for large t, where q ( s ) ∼ s p when s → ∞ , and p > 1 . The terms of the expansion are defined via a simple iterative algorithm. Using this we show that the box dimension of the related q-clothoid, also called the generalized Euler or Cornu spiral, is equal to d = 2 p / ( 2 p - 1 ) . Furthermore, this curve is Minkowski measurable, and we compute its d-dimensional Minkowski content. We also find oscillatory dimension of Fresnel integrals by studying the corresponding chirps.

Box dimension of spiral trajectories of some vector fields in ℝ3

We study the behaviour of Minkowski content of bounded sets under bi-Lipschitzian mappings. Applications include Minkowski contents and box dimensions of spirals in ℝ3, dynamical systems, and singular integrals.

BOX DIMENSION AND MINKOWSKI CONTENT OF THE CLOTHOID

We prove that the box dimension of the standard clothoid is equal to d = 4/3. Furthermore, this curve is Minkowski measurable, and we compute its d-dimensional Minkowski content. Oscillatory dimensions of component functions of the clothoid are also equal to 4/3.

Multiplicity of fixed points and growth of ε-neighborhoods of orbits

Abstract We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on e of the length of e-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered in Elezovic, Žubrinic and Županovic (2007) [5] in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity of fixed points and the dependence on e of the length of e-neighborhoods of orbits in non-differentiable cases. Applications include in particular Poincare maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the e-neighborhood of one orbit of the Poincare map (for example numerically), and by comparing it to the appropriate scale.

Fractal properties of Bessel functions

A fractal oscillatority of solutions of second-order differential equations near infinity is measured by oscillatory and phase dimensions. The phase dimension is defined as a box dimension of the trajectory

Fractal analysis of Hopf bifurcation at infinity

(x,\dot{x})

Fractal analysis of spiral trajectories of some planar vector fields

in

Poincaré map in fractal analysis of spiral trajectories of planar vector fields

\mathbb{R}^2

Box dimension of trajectories of some discrete dynamical systems

of a solution