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Zbl 1087.37048
Berti, Massimiliano; Bolle, Philippe
A functional analysis approach to Arnold diffusion. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, No. 4, 395-450 (2002). ISSN 0294-1449

This paper is devoted to the study of instabilities in Hamiltonian near-integrable systems in the particular case of the quasi-periodically forced pendulum. The system is given by the Hamiltonian function $$ \Cal H_\mu =\omega \cdot I +\frac{p^2}{2}+(\cos q -1)+\mu f(\varphi, q), $$ where $(\varphi, q)\in \Bbb T^n \times {\Bbb T}^1$ are the angle variables, $(I,p)\in {\Bbb T}^n \times {\Bbb T}^1$ are the action variables, $\mu\geq 0$ is a small real parameter, and $\omega\in \Bbb T^n$ is the frequency vector. The corresponding system can be written as $$ -\ddot q +\sin q =\mu\sin q\ f(\omega t+A, q),$$ where $A\in \Bbb R^n$ is a constant. The main result of the paper is the following theorem: Suppose that the frequency vector $\omega$ is Diophantine, and the Poincaré-Melnikov function $$ M(A) = \int_{\Bbb R} [f(\omega t +A, 0)-f(\omega t +A ,q_0(t))]dt, $$ with $q_0(t)=4\arctan(\exp t)$ being the unperturbed separatrix of the pendulum, posesses a proper minimum or maximum. Then, for $\mu$ small enough, there are orbits whose action variables $I$ undergo a drift of order one under the time interval O$((1/\mu)\log (1/\mu))$. It was known before that in initially unstable systems, of which the above system is an example, instability can develop over a polynomial time. The above theorem gives the best known estimate on the speed of diffusion. The authors give a detailed proof for the case when $f$ is independent of $q$, and explain the modifications needed for the general case.
[Maria Saprykina (Toronto)]

MSC 2000:: *37J40 Perturbations, etc.
37J45 Periodic, homoclinic and heteroclinic orbits, etc.

Keywords: Arnold diffusion; shadowing theorem; splitting of separatrices; heteroclinic orbits; variational methods; nonlinear functional analysis

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