By Tassos Bountis

Creation -- Hamiltonian platforms of Few levels of Freedom -- neighborhood and international balance of movement -- common Modes, Symmetries and balance -- effective symptoms of Ordered and Chaotic movement -- FPU Recurrences and the Transition from susceptible to robust Chaos -- Localization and Diffusion in Nonlinear One-Dimensional Lattices -- The Statistical Mechanics of Quasi-stationary States -- Conclusions, Open difficulties and destiny Outlook

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For this reason points A and B in Fig. 0; 0/ fixed point, which is called stable. More precisely, in view of the theory presented in Chap. 0; 0/ is characterized by what we called conditional (or neutral) stability. Let us now discuss the linear stability of fixed points of a Hamiltonian system within the more general framework outlined in Sect. 2. 11) k D N C 1; : : : ; 2N; N where xj represents small variations away from the origin (qj D pj D 0, j D 1; : : : ; N ) of the qj variables for j D 1; : : : ; N and the pj variables for j D N C 1; : : : ; 2N .

In fact, all indications for the H´enon-Heiles system is that a second integral for all A, B, C does not exist. ”, you may rightfully ask. Well, this is a very good question. A first answer may be given by the fact that in many cases that have been integrated numerically by many authors, even close to the equilibrium point (i), one finds, besides periodic and quasiperiodic orbits, a new kind of solution that appears “irregular” and “unpredictable”, one may even say random-looking [226]! These are the orbits we have called chaotic.

It does not tell us how to find the N integrals that must exist (according to the LA theorem) and which are necessary to solve the equations of motion by quadratures. But do we really want to get into all this trouble? First of all the integrals are not at all easy to find. Even for the simple-looking H´enon-Heiles system , the best one can do is assume that the second integral F is a polynomial in the momentum and position variables and solve for its coefficients by direct differentiation. What happens, however, if F turns out to be an infinite series?

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