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March  2004, 11(2&3): 413-448. doi: 10.3934/dcds.2004.11.413

On the double pendulum: An example of double resonant situations

Antonio Pumariño 1, and  Claudia Valls 2,

1. 

Departamento de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, Spain
2. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal

Received  July 2002 Revised  January 2004 Published  June 2004

As a model of double resonant situations, we study fast periodic perturbations of a double pendulum. The associated dynamical system presents periodic orbits whose invariant manifolds split under the perturbation. The main purpose of this paper is to analytically show that this splitting is given, in first order, by the Melnikov function and give a lower bound for such splitting in terms of the perturbative parameter. Many results used in "simple pendulum cases" have to be adapted in order to give a description of the intricate dynamics exhibited by these periodic perturbations.
Keywords: A double pendulum, invariant manifolds, by the Melnikov function..
Mathematics Subject Classification: 37J40, 34C29, 34C37, 37G2.
Citation: Antonio Pumariño, Claudia Valls. On the double pendulum: An example of double resonant situations. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 413-448. doi: 10.3934/dcds.2004.11.413
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