Renormalization method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems

An approximate renormalization procedure is derived for the HamiltonianH(v,x,t)=v2/2−M cosx−P cosk(x−t). It gives an estimate of the large scale stochastic instability threshold which agrees within 5–10% with the results obtained from direct numerical integration of the canonical equations. It shows that this instability is related to the destruction of KAM tori between the two resonances and makes the connection with KAM theory. Possible improvements of the method are proposed. The results obtained forH allow us to estimate the threshold for a large class of Hamiltonian systems with two degrees of freedom.