19 research outputs found
Fractal analysis of hyperbolic saddles with applications
In this paper we express the Minkowski dimension of spiral trajectories near
hyperbolic saddles and semi-hyperbolic singularities in terms of the Minkowski
dimension of intersections of such spirals with transversals near these
singularities. We apply these results to hyperbolic saddle-loops and hyperbolic
-cycles to obtain upper bounds on the cyclicity of such limit periodic sets.Comment: 16 pages, 2 figure
Invariance of the normalized Minkowski content with respect to the ambient space
It is easy to show that the lower and the upper box dimensions of a bounded
set in Euclidean space are invariant with respect to the ambient space. In this
article we show that the Minkowski content of a Minkowski measurable set is
also invariant with respect to the ambient space when normalized by an
appropriate constant. In other words, the value of the normalized Minkowski
content of a bounded, Minkowski measurable set is intrinsic to the set.Comment: 11 pages, 0 figure
Multiplicity of fixed points and growth of epsilon-neighbourhoods of orbits
We study the relationship between the multiplicity of a fixed point of a
function g, and the dependence on epsilon of the length of epsilon-neighborhood
of any orbit of g, tending to the fixed point. The relationship between these
two notions was discovered before (Elezovic, Zubrinic, Zupanovic) 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 epsilon of the
length of epsilon-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 epsilon-neighborhood of one orbit
of the Poincare map (for example numerically), and by comparing it to the
appropriate scale.Comment: 29 pages, 2 figures, to appear in Journal of Differential Equation