Fractal analysis of spiral trajectories of some vector fields in $ {\mathbb{R}}^{3}$

Darko Žubrinić; Vesna Županović

doi:10.1016/j.crma.2006.04.021

Dynamical Systems

Fractal analysis of spiral trajectories of some vector fields in

R^{3}

[Analyse fractale des trajectoires spirales de quelques champs de vecteurs dans

R^{3}

]

Présenté par : Étienne Ghys

Darko Žubrinić ¹ ; Vesna Županović ¹

¹ University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia

Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 959-963.

Résumé
Abstract

Nous étudions la ‘box dimension’ et le contenu de Minkowski des solutions spirales de quelques systèmes dynamiques dans $R^{3}$ .

We study box dimension and Minkowski content of spiral solutions of some dynamical systems in $R^{3}$ .

Reçu le : 2005-03-02
Accepté le : 2006-03-30
Publié le : 2006-05-11

DOI : 10.1016/j.crma.2006.04.021

Affiliations des auteurs :

Darko Žubrinić ¹ ; Vesna Županović ¹

¹ University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia

@article{CRMATH_2006__342_12_959_0,
     author = {Darko \v{Z}ubrini\'c and Vesna \v{Z}upanovi\'c},
     title = {Fractal analysis of spiral trajectories of some vector fields in $ {\mathbb{R}}^{3}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {959--963},
     publisher = {Elsevier},
     volume = {342},
     number = {12},
     year = {2006},
     doi = {10.1016/j.crma.2006.04.021},
     language = {en},
}

TY  - JOUR
AU  - Darko Žubrinić
AU  - Vesna Županović
TI  - Fractal analysis of spiral trajectories of some vector fields in $ {\mathbb{R}}^{3}$
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 959
EP  - 963
VL  - 342
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2006.04.021
LA  - en
ID  - CRMATH_2006__342_12_959_0
ER  -

%0 Journal Article
%A Darko Žubrinić
%A Vesna Županović
%T Fractal analysis of spiral trajectories of some vector fields in $ {\mathbb{R}}^{3}$
%J Comptes Rendus. Mathématique
%D 2006
%P 959-963
%V 342
%N 12
%I Elsevier
%R 10.1016/j.crma.2006.04.021
%G en
%F CRMATH_2006__342_12_959_0

Darko Žubrinić; Vesna Županović. Fractal analysis of spiral trajectories of some vector fields in $ {\mathbb{R}}^{3}$. Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 959-963. doi : 10.1016/j.crma.2006.04.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.021/

Bibliographie
Cité par

[1] M. Caubergh; F. Dumortier Hopf–Takens bifurcations and centers, J. Differential Equations, Volume 202 (2004) no. 1, pp. 1-31

[2] M. Caubergh; J.P. Françoise Generalized Liénard equations, cyclicity and Hopf–Takens bifurcations, Qualitative Theory of Dynamical Systems, Volume 6 (2005), pp. 195-222

[3] Y. Dupain; M. Mendès France; C. Tricot Dimension de spirales, Bull. Soc. Math. France, Volume 111 (1983), pp. 193-201

[4] L.C. Evans; R.F. Gariepy Measure Theory and Fine Properties of Functions, CRC Press, 1992

[5] K. Falconer Fractal Geometry, John Wiley and Sons, 1990

[6] A. Gray Tubes, Addison-Wesley, 1990

[7] J. Guckenheimer; P. Holmes Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983

[8] C.Q. He; M.L. Lapidus Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function, Mem. Amer. Math. Soc., Volume 127 (1997) no. 608

[9] L. Horvat; D. Žubrinić Maximally singular Sobolev functions, J. Math. Anal. Appl., Volume 304 (2005) no. 2, pp. 531-541

[10] M.L. Lapidus; C. Pomerance The Riemann zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums, Proc. London Math. Soc. (3), Volume 66 (1993) no. 1, pp. 41-69

[11] P. Mattila Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Univ. Press, Cambridge, 1995

[12] M. Pašić Minkowski–Bouligand dimension of solutions of the one-dimensional p-Laplacian, J. Differential Equations, Volume 190 (2003), pp. 268-305

[13] M. Pašić; V. Županović Some metric-singular properties of the graph of solutions of the one-dimensional p-Laplacian, Electronic J. Differential Equations, Volume 2004 (2004) no. 60, pp. 1-25

[14] F. Takens Unfoldings of certain singularities of vector fields: Generalized Hopf bifurcations, J. Differential Equations, Volume 14 (1973), pp. 476-493

[15] C. Tricot Curves and Fractal Dimension, Springer-Verlag, 1995

[16] D. Žubrinić Singular sets of Sobolev functions, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 539-544

[17] D. Žubrinić Singular sets of Lebesgue integrable functions, Chaos, Solitons, Fractals, Volume 21 (2004), pp. 1281-1287

[18] D. Žubrinić, Analysis of Minkowski contents of fractal sets and applications Real Anal. Exchange, in press

[19] D. Žubrinić; V. Županović Fractal analysis of spiral trajectories of some planar vector fields, Bull. Sci. Math., Volume 129 (2005) no. 6, pp. 457-485

[20] D. Žubrinić, V. Županović, Box dimension of spiral trajectories of some vector fields in $R^{3}$ , preprint

Renato Huzak; Kristian Uldall Kristiansen; Goran Radunovic Slow divergence integral in regularized piecewise smooth systems, Electronic Journal of Qualitative Theory of Differential Equations (2024) no. 15, p. 1 | DOI:10.14232/ejqtde.2024.1.15
Renato Huzak; Domagoj Vlah; Darko Žubrinić; Vesna Županović Fractal analysis of degenerate spiral trajectories of a class of ordinary differential equations, Applied Mathematics and Computation, Volume 438 (2023), p. 127569 | DOI:10.1016/j.amc.2022.127569
Peter De Maesschalck; Renato Huzak; Ansfried Janssens; Goran Radunović Minkowski Dimension and Slow–Fast Polynomial Liénard Equations Near Infinity, Qualitative Theory of Dynamical Systems, Volume 22 (2023) no. 4 | DOI:10.1007/s12346-023-00854-4
L. Korkut; D. Vlah; V. Županović Fractal properties of Bessel functions, Applied Mathematics and Computation, Volume 283 (2016), p. 55 | DOI:10.1016/j.amc.2016.02.025
Lana Horvat Dmitrović Box dimension of Neimark–Sacker bifurcation, Journal of Difference Equations and Applications, Volume 20 (2014) no. 7, p. 1033 | DOI:10.1080/10236198.2014.884085
Lana Horvat Dmitrović Box dimension and bifurcations of one-dimensional discrete dynamical systems, Discrete Continuous Dynamical Systems - A, Volume 32 (2012) no. 4, p. 1287 | DOI:10.3934/dcds.2012.32.1287
Mervan Pašić; Darko Žubrinić; Vesna Županović Oscillatory and phase dimensions of solutions of some second-order differential equations, Bulletin des Sciences Mathématiques, Volume 133 (2009) no. 8, p. 859 | DOI:10.1016/j.bulsci.2008.03.004
L. KORKUT; D. ŽUBRINIĆ; V. ŽUPANOVIĆ BOX DIMENSION AND MINKOWSKI CONTENT OF THE CLOTHOID, Fractals, Volume 17 (2009) no. 04, p. 485 | DOI:10.1142/s0218348x09004570
Luka Korkut; Domagoj Vlah; Darko Žubrinić; Vesna Županović Generalized Fresnel integrals and fractal properties of related spirals, Applied Mathematics and Computation, Volume 206 (2008) no. 1, p. 236 | DOI:10.1016/j.amc.2008.09.009
Darko Žubrinić; Vesna Županović Box dimension of spiral trajectories of some vector fields in ℝ3, Qualitative Theory of Dynamical Systems, Volume 6 (2005) no. 2, p. 251 | DOI:10.1007/bf02972676

Cité par 10 documents. Sources : Crossref

Commentaires - Politique