Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space (Q1576771)

The Hamiltonian \( {1\over 2} (I_1^2+I_2^2)+\varepsilon(\cos \phi_1-1) (1+\mu(\sin\phi_2+\cos t))\), \(I \in {\mathbb{R}}^2\), is considered like in Arnold's model about diffusion. Using the fixed point theorems, one proves the existence of the stable and unstable manifolds of invariant, ``a priori unstable tori'', for any vector-frequency \((\omega,1)\in {\mathbb{R}}^2\). The author provides a detailed proof of the content of Assertion B of \textit{V. I. Arnold}'s paper [``Instabilities of dynamical systems with several degrees of freedom'', Sov. Math., Dokl. 5, 581-585 (1964); translation from Dokl. Akad. Nauk SSSR 156, 9-12 (1964; Zbl 0135.42602)]. The proofs are based on technical tools suggested by Arnold, i.e. the contraction mapping method together with the ``conical metric''.