Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space

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April 1998, 4(2): 379-391. doi: 10.3934/dcds.1998.4.379

Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space

1.	Dipartimento di matematica II Università di Roma, via della Ricerca Scientifica 00133 Roma

Received August 1997 Published February 1998

Abstract
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We consider the hamiltonian $H=1/2(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1) (1+\mu(\sin\varphi_2+\cos t))$ $I\in\mathbb{R}^2$ ("Arnol'd model about diffusion"); by means of fixed point theorems, the existence of the stable and unstable manifolds (whiskers) of invariant, "a priori unstable tori", for any vector-frequency $(\omega,1)\in\mathbb{R}^2$ is proven. Our aim is to provide detailed proofs which are missing in Arnol'd's paper, namely prove the content of the Assertion B pag.583 of [A]. Our proofs are based on technical tools suggested by Arnol'd i.e. the contraction mapping method together with the "conical metric" (see the footnote ** of pag. 583 of [A]).

Keywords: Fixed point theorems, Arnol'd model..

Mathematics Subject Classification: 37C2.

Citation: Paolo Perfetti. Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 379-391. doi: 10.3934/dcds.1998.4.379

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