Comptes Rendus
no. 12
Dynamical Systems
Fractal analysis of spiral trajectories of some vector fields in R3
[Analyse fractale des trajectoires spirales de quelques champs de vecteurs dans R3]

Présenté par : Étienne Ghys

Darko Žubrinić1 ; Vesna Županović1
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.

Nous étudions la ‘box dimension’ et le contenu de Minkowski des solutions spirales de quelques systèmes dynamiques dans R3.

We study box dimension and Minkowski content of spiral solutions of some dynamical systems in R3.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.04.021
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/

[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 R3, preprint

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Singular sets of Sobolev functions

Darko Žubrinić

C. R. Math (2002)

Seuils d'instabilités des écoulements de type rotor–stator avec flux axial

Sébastien Poncet; Marie-Pierre Chauve

C. R. Méca (2006)