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Mathematical Problems in Engineering
Volume 2011, Article ID 723629, 11 pages
http://dx.doi.org/10.1155/2011/723629
Research Article

Numerical Integration and Synchronization for the 3-Dimensional Metriplectic Volterra System

1Seminarul de Geometrie şi Topologie, West University of Timişoara, 4, B-dul V. Pârvan, 300223 Timişoara, Romania
2Math Department, The “Politehnica” University of Timişoara, Piaţa Victoriei nr. 2, 300006 Timişoara, Romania

Received 21 January 2010; Accepted 29 June 2010

Academic Editor: Marcelo Messias

Copyright © 2011 Gheorghe Ivan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. N. Kaufman, “Dissipative Hamiltonian systems: a unifying principle,” Physics Letters A, vol. 100, no. 8, pp. 419–422, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  2. P. Birtea, M. Boleanţu, M. Puta, and R. M. Tudoran, “Asymptotic stability for a class of metriplectic systems,” Journal of Mathematical Physics, vol. 48, no. 8, Article ID 082703, 7 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Grmela, “Bracket formulation of dissipative fluid mechanics,” Physics Letters A, vol. 102, no. 8, pp. 355–358, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  4. P. J. Morrison, “A paradigm for joined Hamiltonian and dissipative systems,” Physica D, vol. 18, no. 1–3, pp. 410–419, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. Fish, Geometrical structures of the metriplectic dynamical systems, Ph.D. thesis, Portland, Ore, USA, 2005.
  6. P. Guha, “Metriplectic structure, Leibniz dynamics and dissipative systems,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 121–136, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. G. Ivan and D. Opriş, “Dynamical systems on Leibniz algebroids,” Differential Geometry—Dynamical Systems, vol. 8, pp. 127–137, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J.-P. Ortega and V. Planas-Bielsa, “Dynamics on Leibniz manifolds,” Journal of Geometry and Physics, vol. 52, no. 1, pp. 1–27, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. N. Agiza and M. T. Yassen, “Synchronization of Rössler and Chen chaotic dynamical systems using active control,” Physics Letters A, vol. 278, no. 4, pp. 191–197, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  10. O. T. Chiş and D. Opriş, “Synchronization and cryptography using chaotic dynamical systems,” in Proceedings of the International Conference of Differential Geometry and Dynamical Systems, vol. 16 of BSG Proceedings [BGSP], pp. 47–56, Geometry Balkan Press, Bucharest, Romania, 2009.
  11. M.-C. Ho, Y.-C. Hung, and I.-M. Jiang, “On the synchronization of uncertain chaotic systems,” Chaos, Solitons & Fractals, vol. 33, pp. 540–546, 2007. View at Publisher · View at Google Scholar
  12. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. D. Fadeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, Germany, 1986.
  14. M. Puta and M. Khashan, “Poisson geometry of the Volterra model,” Analele Universităţii din Timişoara, Seria Matematică—Informatică, vol. 39, pp. 377–384, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  16. W. Kahan, “Unconventional numerical methods for trajectory calculations,” Unpublished lecture notes, 1993.
  17. D. V. Zenkov, A. M. Bloch, and J. E. Marsden, “The energy-momentum method for the stability of non-holonomic systems,” Dynamics and Stability of Systems, vol. 13, no. 2, pp. 123–165, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. B. Suris, “Integrable discretizations for lattice system: local equations of motion and their Hamiltonian properties,” Reviews in Mathematical Physics, vol. 11, no. 6, pp. 727–822, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet