We consider the hamiltonian $H={1\over2}(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1) (1+\mu(\sin\varphi_2+\cos t))$ $I\in{\Bbb R}^2$ (\lq\lq Arnol'd model about diffusion"); by means of fixed point theorems, the existence of the stable and unstable manifolds {\it (whiskers)} of invariant, \lq\lq a priori unstable tori", for any vector-frequency $(\omega,1)\in{\Bbb R}^2$ is proven. Our aim is to provide detailed proofs which are missing in Arnol'd's paper, namely prove the content of the {\tt Assertion B} pag.583 of [A]. Our proofs are based on technical tools suggested by Arnol'd i.e. the contraction mapping method togheter with the \lq\lq conical metric" ( see the footnote ** of pag. 583 of [A]).
Perfetti, P. (1998). Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 4(2), 379-391.
Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space
PERFETTI, PAOLO
1998-04-01
Abstract
We consider the hamiltonian $H={1\over2}(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1) (1+\mu(\sin\varphi_2+\cos t))$ $I\in{\Bbb R}^2$ (\lq\lq Arnol'd model about diffusion"); by means of fixed point theorems, the existence of the stable and unstable manifolds {\it (whiskers)} of invariant, \lq\lq a priori unstable tori", for any vector-frequency $(\omega,1)\in{\Bbb R}^2$ is proven. Our aim is to provide detailed proofs which are missing in Arnol'd's paper, namely prove the content of the {\tt Assertion B} pag.583 of [A]. Our proofs are based on technical tools suggested by Arnol'd i.e. the contraction mapping method togheter with the \lq\lq conical metric" ( see the footnote ** of pag. 583 of [A]).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
