Complex dynamics and chaos in commutable pendulum
V.R. Nosov, H. Dominguez, J.A. Ortega–Herrera, and J.A. Meda–Campaña
Instituto Politécnico Nacional, Department of Mechanical Engineering, SEPI–ESIME Zacatenco del IPN, México D.F., México, Phone: (052) (55) 57296000 Ext. 54737 e–mail: nosovval@hotmail.com; hdoma@hotmail.com; jmedac@ipn.mx
Recibido el 13 de diciembre de 2010.
Aceptado el 23 de noviembre de 2011.
Abstract
This paper deals with a commutable pendulum which has two different natural frequencies in two space regions. The first region corresponds to the first three quadrants of the phase space; while the second region coincides with the fourth quadrant. This apparently trivial system shows a very complex behavior, regardless of the fact that it is based on the simple pendulum model. In the case where the two natural frequencies coincide, the flow is stable. However, just by varying the natural frequency in the fourth quadrant, the system may be asymptotically stable or unstable and also may have simple limit cycle, complex limit cycles and even chaotic behavior. In all these cases the trajectories have simple analytical descriptions.
Keywords: Chaos; commutable pendulum; complex dynamical system.
Resumen
En este trabajo se estudia el péndulo con conmutaciones de frecuencia en dos regiones del plano. La primera región coincide con los tres primeros cuadrantes del plano, mientras que la segunda región coincide con el cuarto cuadrante. Este sistema, aparentemente trivial, presenta un comportamiento realmente complejo a pesar de estar basado en el modelo del péndulo simple. En situaciones cuando las dos frecuencias naturales coinciden, el comportamiento es estable. Sin embargo, cuando se cambia la frecuencia natural en el cuarto cuadrante, el sistema puede ser asintóticamente estable o inestable, puede tener un ciclo límite sencillo, ciclos límites complejos y más aún, puede presentar comportamiento caótico. Otra característica importante es que en todos estos casos las trayectorias pueden ser descritas analíticamente de forma sencilla.
Descriptores: Caos; péndulo conmutable; sistema dinámico complejo.
PACS: 05.45.–a; 05.45.Ac
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Acknowledgments
The authors gratefully acknowledge the support of CONACYT through SNI scholarship, and of IPN through research projects 20110250 and 20110812, as well as scholarships EDI and COFAA.
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