Luka Korkut, Domagoj Vlah, Vesna Zupanovic
Abstract:
We study a class of second-order nonautonomous differential equations,
and the corresponding planar and spatial systems, from the geometrical
point of view. The oscillatory behavior of solutions at infinity is
measured by oscillatory and phase dimensions,
The oscillatory dimension is defined as the box dimension of the
reflected solution near the origin, while the phase dimension is
defined as the box dimension of a trajectory of the planar system
in the phase plane. Using the phase dimension of
the second-order equation we compute the box dimension of a spiral
trajectory of the spatial system. This phase dimension of the second-order
equation is connected to the asymptotic of the associated Poincare map.
Also, the box dimension of a trajectory of the reduced normal form with
one eigenvalue equals zero, and a pair of pure imaginary eigenvalues is
computed when limit cycles bifurcate from the origin.
Submitted September 28, 2015. Published October 29, 2015.
Math Subject Classifications: 37C45, 37G10, 34C15, 28A80.
Key Words: Spiral; chirp; box dimension; rectifiability; oscillatory dimension;
phase dimension; limit cycle.
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Luka Korkut University of Zagreb Faculty of Electrical Engineering and Computing Department of Applied Mathematics, Unska 3 10000 Zagreb, Croatia email: luka.korkut@fer.hr |
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Domagoj Vlah University of Zagreb Faculty of Electrical Engineering and Computing Department of Applied Mathematics, Unska 3 10000 Zagreb, Croatia email: domagoj.vlah@fer.hr |
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Vesna Zupanovic University of Zagreb Faculty of Electrical Engineering and Computing Department of Applied Mathematics, Unska 3 10000 Zagreb, Croatia email: vesna.zupanovic@fer.hr |
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