Hamilton-Poisson realizations for the Lü system. (English) Zbl 1223.34067

Summary: The Hamilton-Poisson geometry has proved to be an interesting approach for a lot of dynamics arising from different areas like biology, economics, or engineering. The Lü system was first proposed by Lü and Chen as a model of a nonlinear electrical circuit, and it was studied from various points of view. We study it from the mechanical geometry point of view and point out some of its geometrical and dynamical properties.


34C28 Complex behavior and chaotic systems of ordinary differential equations
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[1] H. Gümral and Y. Nutku, “Poisson structure of dynamical systems with three degrees of freedom,” Journal of Mathematical Physics, vol. 34, no. 12, pp. 5691-5723, 1993. · Zbl 0783.58036
[2] C. D\uan\uaias\ua, C. Hedrea, C. Pop, and M. Puta, “Some geometrical aspects in the theory of Lagrange system,” Tensor, New Series, vol. 69, pp. 83-87, 2008. · Zbl 1193.37069
[3] J.-M. Ginoux and B. Rossetto, “Differential geometry and mechanics: applications to chaotic dynamical systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp. 887-910, 2006. · Zbl 1111.37021
[4] J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 3, pp. 659-661, 2002. · Zbl 1063.34510
[5] F. Haas and J. Goedert, “On the generalized Hamiltonian structure of 3D dynamical systems,” Physics Letters A, vol. 199, no. 3-4, pp. 173-179, 1995. · Zbl 1020.35533
[6] B. Hernández-Bermejo and V. Fairén, “Simple evaluation of Casimir invariants in finite-dimensional Poisson systems,” Physics Letters A, vol. 241, no. 3, pp. 148-154, 1998. · Zbl 0945.70517
[7] M. W. Hirsch, S. Smale, and R. L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Elsevier, New York, NY, USA, 2003. · Zbl 1239.37001
[8] P. Birtea and M. Puta, “Equivalence of energy methods in stability theory,” Journal of Mathematical Physics, vol. 48, no. 4, pp. 81-99, 2007. · Zbl 1137.34338
[9] P. Birtea, M. Puta, and R. M. Tudoran, “Periodic orbits in the case of a zero eigenvalue,” Comptes Rendus Mathématique, vol. 344, no. 12, pp. 779-784, 2007. · Zbl 1131.34034
[10] C. Pop, I. David, and A. I. Popescu-Busan, “A new approach about Lu system, advanced in mathematical and computational methods,” in Proceedings of the 12th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, pp. 277-281, Faro, Portugal, November 2010.
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