ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Nikiforov A. I., Ryabov D. S., Chechin G. M. Dynamical chaos in three-dimensional dissipative system with D_2-symmetry group. Izvestiya VUZ. Applied Nonlinear Dynamics, 2004, vol. 12, iss. 6, pp. 28-43. DOI: 10.18500/0869-6632-2004-12-6-28-43

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Language: 
Russian
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
512.54; 517.925.42
DOI: 
10.18500/0869-6632-2004-12-6-28-43

Dynamical chaos in three-dimensional dissipative system with D_2-symmetry group

Autors: 
Nikiforov Anton Igorevich, Brigham Young University
Ryabov Denis Sergeevich, Southern Federal University
Chechin Georgij Mihajlovich, Southern Federal University
Abstract: 

Earlier, we found all three-dimensional dissipative systems with quadratic nonlinearities which are invariant with respect to any of 32 point symmetry groups of crystallographic symmetry and which can demonstrate chaotic behavior for appropriate values оf their pertinent parameters. Among these systems, there is one with е D_2-symmetry group. This system seems to be more simple and more elegant than those by Lorenz and Rossler which also belong to the above mentioned class of dissipative systems. In the present paper, we investigate regular and chaotic dynamics оf the Dy-symmetry system emphasizing the usefulness of the general theory of the nonlinear dynamical systems with discrete symmetries.

Key words: 
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Acknowledgments: 
We express our sincere gratitude to V.P. Sakhnenko for support of the research direction developed by the authors and to N.A. Kolodin for carrying out a number of calculations at the initial stage of preparation of this article.
Reference: 
  1. Sprott JC. Sоmе simple chaotic flows. Phys. Rev. Е. 1994;50(2):647. DOI:  10.1103/PhysRevE.50.R647.
  2. Lorenz EN. Deterministic nonperiodic flow. J. Atmos. Sci. 1963;20(2):130–141. DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
  3. Rossler OE. An equation for continuous chaos. Phys. Lett. А. 1976;57(5):397–398. DOI:10.1016/0375-9601(76)90101-8.
  4. Chechin GM, Ryabov DS. Three-dimensional chaotic flows with discrete symmetries. Phys. Rev. Е. 2004;69(3):036202. DOI:  10.1103/PhysRevE.69.036202.
  5. Sakhnenko VP, Chechin GM. Symmetry selection rules in nonlinear dynamics of atomic displacements. Dokl. of Acad. of Sciences. 1993;330:308–310. (in Russian).
  6. Chechin GM, Sakhnenko VP. Interaction between normal modes 10 nonlinear dynamical systems with discrete symmetry. Exact results. Physica D. 1998;117(1-4):43–76. DOI: 10.1016/S0167-2789(98)80012-2.
  7. Kuznetsov SP. Dynamic Chaos. (Series "Modern Theory of Vibrations and Waves"). Moskow: Fizmatlit; 2001. 295 p.
  8. Magnitsky NA, Sidorov CB. New Methods of Chaotic Dynamics. Moscow: Editorial  URSS; 2004. 318 p.
  9. Gilmore R. Topological analysis of chaotic dynamical systems. Rev. Mod. Phys. 1998;70(4):1455-1530. DOI: 10.1103/RevModPhys.70.1455.
  10. Letellier C, Gilmore R. Covering dynamical systems: twofold covers. Phys. Rev. E. 2003;63(1):016206. DOI: 10.1103/PhysRevE.63.016206.
Received: 
16.11.2004
Accepted: 
02.03.2005
Published: 
15.06.2005
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