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

Žubrinić, Darko; Županović, Vesna

doi:10.36045/bbms/1228486418

Cited by 34 publications

(54 citation statements)

References 3 publications

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“…Going through the proof of [16,Lemma 4], it can be computed that the principal part for the nucleus is constant and equal to (20) H…”

Section: Properties Of the Principal Parts Of Areasmentioning

confidence: 99%

ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

Resman¹

2014

Nonlinearity

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In this article, we study analyticity properties of (directed) areas of ε-neighborhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using ε-neighborhoods of orbits in the simplest formal class.We show that the coefficient in front of ε 2 term in the asymptotic expansion in ε, which we call the principal part of the area, is a sectorially analytic function in the initial point of the orbit. It satisfies a cohomological equation similar to the standard trivialization equation for parabolic diffeomorphisms.We give necessary and sufficient conditions on a diffeomorphism f for the existence of globally analytic solution of this equation. Furthermore, we introduce new classification type for diffeomorphisms implied by this new equation and investigate the relative position of its classes with respect to the analytic classes.

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“…Going through the proof of [16,Lemma 4], it can be computed that the principal part for the nucleus is constant and equal to (20) H…”

Section: Properties Of the Principal Parts Of Areasmentioning

confidence: 99%

ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

Resman¹

2014

Nonlinearity

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“…Also the nondegenerate Hopf bifurcation occurs from the weak focus whose spiral trajectories have box dimension equal to 4/3, also seeŽubrinić anď Zupanović. 8 An interesting singular integral can be generated by the clothoid. Assume that Ω is a bounded open set in the plane containing the clothoid Γ described by (1).…”

Section: Discussionmentioning

confidence: 98%

“…8 It is proved that the box dimensions of spiral trajectories tending to the focus are contained in the discrete set { 4k 2k+1 : k ∈ N}. An interesting fact is that box dimension of the clothoid coincides with the smallest value from that set.…”

Section: Discussionmentioning

confidence: 99%

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Box Dimension and Minkowski Content of the Clothoid

2009

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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. Fractals 2009.17:485-492. Downloaded from www.worldscientific.com by MCGILL UNIVERSITY on 02/05/15. For personal use only. Fractals 2009.17:485-492. Downloaded from www.worldscientific.com by MCGILL UNIVERSITY on 02/05/15. For personal use only. Theorem 4. (Minkowski content under bi-Lipschitz mappings) Let Ω and Ω be open sets in R N , and let H : Ω → Ω be a bi-Lipschitz mapping with lower and upper Lipschitz constants equal to C and C respectively, and its Jacobian defined by J H (x) := det H (x) a.e. Let A be a bounded set such that A ⊆ Ω. Then for all s ≥ 0 we have

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“…In the case of dynamical systems, fractal analysis consists of studying the box dimension and Minkowski content of trajectories or orbits. Several articles with fractal analysis of bifurcations of dynamical systems (see [8], [31], [28], [29]) showed that there is a direct connection between the change in box dimension of trajectories of dynamical systems and the bifurcation of that system. Around the hyperbolic singularities the box dimension is trivial (0), while around the nonhyperbolic singularities the box dimension is positive and connected to the appropriate bifurcation.…”

Section: Introductionmentioning

confidence: 99%

Box dimension of Neimark–Sacker bifurcation

Dmitrović

2014

Journal of Difference Equations and Applications

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In this paper we show how a change of box dimension of the orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a nonhyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems (see [12],[8]). Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected with the type of bifurcation. First, we study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and get the result for the box dimension of the orbit on the center manifold. Then we consider the planar discrete system undergoing a Neimark-Sacker bifurcation. It is shown that box dimension depends on the order of the nondegeneracy at the nonhyperbolic fixed point and on the angle-displacement map. As it was expected, we prove that the box dimension is different in rational and irrational case.

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

Cited by 34 publications

References 3 publications

ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

Box Dimension and Minkowski Content of the Clothoid

Box dimension of Neimark–Sacker bifurcation

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