Palabras clave
center manifold theorem
Hopf bifurcation theorem
sistema tipo Lorenz
teorema de la variedad central
teorema de la bifurcación de Hopf
Cómo citar
Resumen
En este artículo se hace un análisis de la bifurcación de Hopf del sistema tridimensional tipo Lorenz introducido por Xianyi Li y Qianjun Ou (2011), este análisis consiste en identificar una región de parámetros del sistema donde la bifurcación de Hopf es no degenerada y supercrítica, aspecto que no es abordado en el artículo de Xianyi Li y Qianjun Ou. Para lograr este objetivo se utiliza el Teorema de la Variedad Central y el Teorema de Hopf. Además, para ilustrar los resultados, se muestran gráficas de algunas trayectorias del sistema, las cuales fueron obtenidas mediante simulación numérica.
Citas
Algaba, A.; Domínguez, M.; Merino, M.; Rodríguez, A.L. (2015) “Study of the Hopf bifurcation in the Lorenz, Chen and Lü systems", Nonlinear Dyn. 79(1): 885–902.
Algaba, A.; Fernández-Sánchez, F.; Merino, M.; Rodríguez, A.L. (2014) “Centers on center manifolds in the Lorenz, Chen and Lü systems", Commun. Nonlinear Sciences Numerical Simulation. 19(4): 772–775.
Algaba, A.; Fernández-Sánchez, F.; Merino, M.; Rodríguez, A.L. (2013) “The Lü systems is a particular case of the Lorenz system", Physical Letter A. 377(39): 2771–2776.
Castellanos, V.; Blé, G.; Llibre, J. (2016) “Existence of limit cicles in a tritrophic food chain model with holling functional responses of type I and II", Mathematical Methods in the Applied Sciences 39(1): 1–2.
Chen, C.-T. (1994) System and Signal Analysis. Oxford University Press, New York.
Chen, G.; Lü, J. (2003) “Dynamics of the Lorenz family: analysis, control and synchronization", Chinese Science Press, Beijing 00(1): 0–2.
Chen, G.; Lü, J.; Cheng, D. (2004) “A new chaotic system and beyond: The generalized Lorenz-like system", International Journal of Bifurcation and Chaos 14(5): 1507–1537.
Chen, G.; Oi, G. (2005) “Analysis of a new chaotic system", Physical A: Statistical Mechanics and Its Applications 352(2): 295–308.
Dutta, T.K.; Prajapati, P.; Haloi, S. (2016) “Supercritical and subcritical hopf bifurcations in non linear maps", International Journal of Innovative Research in Technology & Science 4(2): 14–20.
Guckenheimer, J.; Holemes, P. (1990) “Nonlinear oscillations dynamical systems and bifurcations of vectors fields”, Ithaca, Fall.
Kuznetsov, Y. A. (2004) Elements of Applied Bifurcation Theory, third edition. Springer-Verlag, New York.
Li, H.; Wang, M. (2013) “Hopf bifurcation analysis in a Lorenz-type systems", Nonlinear Dynamical 71(1): 235–240.
Li, X.; Ou, Q. (2011) “Dynamical properties and simulation of a new Lorenz-like chaotic system", Nonlinear Dynamical 65(3): 255–270.
Liao, X.; Wang, L.; Yu, P. (2007) Stability of Dynamical Systems. London, Canada.
Lorenz, E. (1963) “Deterministic nonperiodic flow", J. Atmos. Sci. 20(2): 130–141.
Pehlivan, I.; Uyaroglu, Y. (2010) “A new chaotic attractor from general Lorenz system family and its electronic experimental implementation", Turk J. Elec. Eng. & Comp. Sci. 18(2): 171–184.
Perko, L. (2000) Differential Equations and Dynamical Systems, third edition. Springer-Verlag, New York.
Pyragas, K. (2006) “Delayed feedback control of chaos", Phil. Trans. R. soc. A. 364(1846): 2309–2334.
Soon Tee, L.; Salleh, Z. (2013) “Dynamical analysis of a modified Lorenz system", Hindawwi Publishing Corporation Journal of Mathematics 2013(1): 1–8.
Sotomayor, J.; Melo, L.; Braga, D. (2006) Stability and Hopf Bifurcation in the Watt Governor Systems. Cornell University Library, orXiv: Matemáticas/0604177
Tian, Y.-P. (2012) Frequency-Domain Analysis and Desing of Distributed Control Systems. Wiley-IEEE Press.
Yan, Z. (2007) “Hopf bifurcation in the Lorenz-type chaotic system", Chaos, Solitons & Fractals 31(5): 1135–1142.
Yu, P.; Lü, J. (2010) “Bifurcation control for a class of Lorenz-like systems", International Journal of Bifurcation and Chaos 21(9): 2647–2664.