Geodesic
Chaotic dynamics of a three-dimensional endomorphism
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 1, pp. 36-50.

Voir la notice de l'article provenant de la source Math-Net.Ru

The present work describes the phases plane bifurcations of some attractors given by a noninvertible three-dimensional map. This study is conducted through the critical manifolds concepts, generalization of critical points and critical lines introduced by Gumowski and Mira [1, 2]. The phase plane shared within two open regions: the first (denoted $Z_{0}$) each point having no real preimage, and the second (denoted  $Z_{2}$) each point having two real preimages. The regions $Z_{0}$, $Z_{2}$ are separated by the critical manifolds, locus of points having two coincident preimages. This requires the visualization of critical manifolds in the three dimensional phases space. And this work also describes the passage of invariant or attractor curves towards weakly chaotic attractors then towards hyper-chaotic attractors via the contact bifurcation through the critical manifolds, which disappear after the contact bifurcation with the its attraction basin boundary.
Keywords: critical manifold, closed invariant curve, weakly chaos
Mots-clés : chaos. 
@article{JSFU_2019_12_1_a2,
     author = {Hacene Gharout and Nourredine Akroune and Abelkadous Taha and Daniele-Fournier Prunaret},
     title = {Chaotic dynamics of a three-dimensional endomorphism},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {36--50},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2019_12_1_a2/}
}
TY  - JOUR
AU  - Hacene Gharout
AU  - Nourredine Akroune
AU  - Abelkadous Taha
AU  - Daniele-Fournier Prunaret
TI  - Chaotic dynamics of a three-dimensional endomorphism
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2019
SP  - 36
EP  - 50
VL  - 12
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2019_12_1_a2/
LA  - en
ID  - JSFU_2019_12_1_a2
ER  - 
%0 Journal Article
%A Hacene Gharout
%A Nourredine Akroune
%A Abelkadous Taha
%A Daniele-Fournier Prunaret
%T Chaotic dynamics of a three-dimensional endomorphism
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2019
%P 36-50
%V 12
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2019_12_1_a2/
%G en
%F JSFU_2019_12_1_a2
Hacene Gharout; Nourredine Akroune; Abelkadous Taha; Daniele-Fournier Prunaret. Chaotic dynamics of a three-dimensional endomorphism. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 1, pp. 36-50. http://geodesic.mathdoc.fr/item/JSFU_2019_12_1_a2/

[1] I. Gumowski, C. Mira, Dynamique chaotique Transformatios ponctuelles Transition Ordre-Désordre, Cepadues Editions, Toulouse, 1980 | MR

[2] C. Mira, Chaotic Dynamics: From the One-dimensional Endomorphism to the Two-Dimensional Diffeomorphism, World Scientific Publishing, Singapore, 1987 | MR | Zbl

[3] C.E. Frouzakis, L. Gardini, I.G. Kevrekidis, G. Millerioux, C. Mira, “On some proporties of invariant sets of two-dimensional noninvertible maps”, International Journal of Bifurcation and Chaos, 7:7 (1997), 1167–1194 | DOI | MR | Zbl

[4] C. Mira, A. Agliari, L. Gardini, “From the box-within-a-box bifurcation structure to the julia set. Part II: Bifurcation routes to different julia sets from an indirect embedding of a quadratic complex map”, International Journal of Bifurcation and Chaos, 19:10 (2009), 3235–3282 | DOI | MR | Zbl

[5] G. Millerioux, C. Mira, “Noninvertible piecewise linear maps applied to chaos synchronization and secure communications”, International Journal of Bifurcation and Chaos, 7:7 (1997), 1617–1634 | DOI | MR | Zbl

[6] E. Mosekilde, Y. Maistrenko, D. Postnov, Chaotic Synchronization: Applications to living systems, World Scientific Series on Nonlinear science, Series A, 42, Singapore, 2001 | MR

[7] J. Xu, D. Fournier-Prunaret, A.K. Taha, P. Chargé, “Chaos generator for secure transmission using a sine map and a RLC series circuit”, Science China, 1:53 (2010), 129–136

[8] J.C. Sprott, “Simplest dissipative chaotic flow”, Physics Letters A, 228 (1997), 271–274 | DOI | MR | Zbl

[9] G.M. Zaslavsky, Chaos in Dynamic Systems, Harwood, Amsterdam, 1985 | MR

[10] G.I. Bischi, L. Mroz, H. Hauser, “Studying basin bifurcations in nonlinear triopoly games by using 3D visualization”, Nonlinear Analysis, Theory, Methods $\$ Applications, 47:8 (2001), 5325–5341 | DOI | MR | Zbl

[11] C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic dynamics in two-dimensional noninvertible maps, World Scientific Publishing, Singapore, 1996 | MR | Zbl

[12] H.E. Nusse, J.A. Yorke, Dynamics: Numerical Explorations, Springer-Verlag, New York, 1998 | MR | Zbl