CPAA
Convergence of Lindstedt series for the non linear wave equation
G. Gentile V. Mastropietro
Communications on Pure & Applied Analysis 2004, 3(3): 509-514 doi: 10.3934/cpaa.2004.3.509
We prove the existence of oscillatory solutions of the nonlinear wave equation, under irrationality conditions stronger than the usual Diophantine one, by perturbative techniques inspired by the Lindstedt series method originally introduced in classical mechanics to study the existence of invariant tori in quasi-integrable Hamiltonian systems.
keywords: tree formalism Lindstedt series method Dirichlet boundary conditions Nonlinear wave equation perturbation theory Diophantine and irrationality conditions periodic solutions
DCDS
Kam theory, Lindstedt series and the stability of the upside-down pendulum
Michele V. Bartuccelli G. Gentile Kyriakos V. Georgiou
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 413-426 doi: 10.3934/dcds.2003.9.413
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

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