Dynamics of a System Associated with a Piecewise Quadratic Family / Dinámica de un sistema asociado a una familia cuadrática a tramos
DOI:
https://doi.org/10.19053/01217488.v7.n2.2016.3951Palabras clave:
Bifurcaciones, caos, estabilidad, órbita periódica, punto fijo, sistema dinámico.Resumen
Abstract
This paper presents a study, both in analytical and numerical form, of a discrete dynamical system associated with a piecewise quadratic family. The orbits of periods one and two are characterized, and their stability is established. The nonsmooth phenomenon known as border collision is present when there is a period doubling. Lyapunov exponents are calculated numerically to determine the presence of chaos in the system.
Resumen
Presenta un estudio analítico y numérico de la dinámica de un sistema discreto asociado a una familia
cuadrática a tramos; se caracterizan las órbitas de período uno y dos, así como su estabilidad; se muestra
la presencia del fenómeno no suave, conocido como bifurcación por colisión de borde cuando ocurre un
doblamiento de período. Se hallaron numéricamente los exponentes de Lyapunov para detectar la presencia de caos en el sistema.
JEL Classification
ArrayDescargas
Referencias
S. Wiggins, “Introduction to Applier Nonlinear Dynamical Systems and Chaos”, Springer-Verlag, Second Edition, 2003.
M. di Bernardo, M.I. Feigin, S.J. Hogan, M.E. Homer, “Local Analysis of C-Bifurcations in n-Dimensional Piecewise Smooth Dynamical Systems”, Chaos, Solitons and Fractals, vol. 10, no. 11, pp. 1881-1908, 1999.
M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, “Piecewise-smooth Dynamical Systems”, Springer Verlag London Limite, 2008.
M. di Bernardo, A. Nordmark, G. Olivar, “Discontinuity-Induced Bifurcations of Equilibria in Piecewise-Smooth and Impacting Dynamical Systems”, Physica D, vol. 237, no. 1, pp. 119-136, 2008. DOI: https://doi.org/10.1016/j.physd.2007.08.008
B. Robert, C. Robert, “Border Collision Bifurcations in a One-Dimensional Piecewise Smooth Map for a PWM Current- Programmed H-Bridge Inverter”, International Journal of Control, vol. 75, no. 16 & 17, pp. 1356-1367, 2002. DOI: https://doi.org/10.1080/0020717021000023771
Y. Do, S.D. Kim, P.S. Kim, “Stability of Fixed Points Placed on the Border in the Piecewise Linear Systems”, Chaos, Solitons and Fractals, vol. 38, pp. 391-399, 2008. DOI: https://doi.org/10.1016/j.chaos.2006.11.022
P. Jain, S. Banerjee, “Border-Collision Bifurcations in One-Dimensional Discontinuous Map”, International Journal of Bifurcation and Chaos, vol. 13, no. 11, pp. 3341-3351, 2003. DOI: https://doi.org/10.1142/S0218127403008533
W. Zheng, “Chaoization and Stabilization of Electric Motor Drives and Their Industrial Applications”, Tesis Doctoral, The University of Hong Kong, 2008.
G. Yuan, S. Banerjee, E. Ott, J.A. Yorke, “Border-Collision Bifurcations in the Buck Converter”, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, vol. 45, no. 7, pp. 707-716, 1998. DOI: https://doi.org/10.1109/81.703837
V. Avrutin, M. Schanz, “Border-Collision Period-Doubling Scenario”, Physical Review E, vol. 70, no. 2, 2004. DOI: https://doi.org/10.1103/PhysRevE.70.026222
M.A. Hassouneh, E.H. Abed, S. Banerjee, Feedback Control of Border Collision Bifurcations in Two-Dimensional Discrete- Time Systems, Reporte Técnico, University of Maryland, 2002. Disponible en: http://hdl.handle.net/1903/6273.
S. Brianzoni, R. Coppier, E. Michetti, “Complex Dynamics in a Growth Model with Corruption in Public Procurement”, Discrete DyDynamics in Nature and Society, vol. 2011, 2011. doi: 10.1155/2011/862396. DOI: https://doi.org/10.1155/2011/862396
B. Routroy, P.S. Dutta, S. Banerjee, Border Collision Bifurcations in n-Dimensional Piecewise Linear Discontinuous Maps,Cornell University Library, 2006. Disponible en: http://arxiv.org/abs/ nlin/0601038.
O. Eriksson, B Brinne, Y. Zhou, J. Bjorkegren, J. Tegnér, Deconstructing the core dynamics from a complex time-lagged regulatory biological circuit The Institution of Engineering and Technology, 2009, doi:10.1049/iet-syb.2007.0028. DOI: https://doi.org/10.1049/iet-syb.2007.0028
Robert L. Devaney, An introduction to Chaotic Dynamical Systems, Second Edition, Addisson Wesley Publishing Company, Inc. 1989.
Y.A. Kuznetsov, Elements of applied bifurcation theory, Springer Verlag, New York, 2004. DOI: https://doi.org/10.1007/978-1-4757-3978-7
D. Gulick, Encounters with Chaos, McGraw- Hill, Inc. 1992.
B. Brogliato, Nonsmooth Mechanics., Springer Verlag, 1999. DOI: https://doi.org/10.1007/978-1-4471-0557-2
S. Banerjee, G.C. Verghese, Nonlinear phenomena in power electronics, Eds. IEEE Press, Piscataway, 2001. DOI: https://doi.org/10.1109/9780470545393