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March  2003, 9(2): 413-426. doi: 10.3934/dcds.2003.9.413

Kam theory, Lindstedt series and the stability of the upside-down pendulum

Michele V. Bartuccelli 1, , G. Gentile 2, and  Kyriakos V. Georgiou 1,

1. 

Department of Mathematics and Statistics, University of Surrey, GU2 7XH, United Kingdom, United Kingdom
2. 

Dipartimento di Matematica, Università di Roma Tre, Roma, I-00146, Italy

Received  October 2001 Revised  March 2002 Published  December 2002

We consider the planar pendulum with support point oscillating in the vertical direction; the upside-down position of the pendulum corresponds to an equilibrium point for the projection of the motion on the pendulum phase space. By using the Lindstedt series method recently developed in literature starting from the pioneering work by Eliasson, we show that such an equilibrium point is stable for a full measure subset of the stability region of the linearized system inside the two-dimensional space of parameters, by proving the persistence of invariant KAM tori for the two-dimensional Hamiltonian system describing the model.
Keywords: Lindstedt series, averaging, KAM theory, stability., vertically driven pendulum, perturbation theory, nonlinear Mathieu's equation, upside-down pendulum.
Mathematics Subject Classification: Primary 34D20, 34C29, 58F10, 58F27, 58F30, 34B30, 34C1.
Citation: Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
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