Hamiltonian dynamical systems

Hamiltonian dynamical system theory is the mathematical framework on which TST rests many textbooks, of various mathematical sophistication, describe this branch of pure/applied mathematics. Some of the various flavors are [20-24]. Very little of this vast information will be needed here, and we shall try to be as self-consistent as possible. 

K. R. Meyer and Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer, Berlin, 1992. 

Hamiltonian Dynamical Systems, Adv. Chem. Phys. Part B 130, 437 (2005). 

Hamiltonian dynamical system, a reprint selection compiled and introduced by R. S. MacKay and J. D. Meiss, Adam Hilger, Bristol, 1987. 

STRUCTURE OF RESONANCES AND TRANSPORT IN MULTIDIMENSIONAL HAMILTONIAN DYNAMICAL SYSTEMS... 

Understanding in global dynamic behavior in Hamiltonian dynamical systems is a fundamental issue in nonlinear dynamics and statistical physics. Besides the applications in various field such as astronomy, plasma physics, and atomic... 

As a simple model for Hamiltonian dynamical system with several degrees of freedom, we have chosen Froeschle map [9-13], given by... 

The Froeschle map could be taken as a coupled system consisting of standard maps. The uncoupled standard map with b = 0, is studied as a prototype of a Poincare map for Hamiltonian dynamics with two degrees of freedom. Similarly, the above Froeschle map could be regarded as a Poincare map of Hamiltonian dynamical system with three degrees of freedom, and it provides a prototype model for such system. 

In the regions where resonances are isolated, the states remain the same regions with high probabilities thus, these regions act as the effective barriers for transitions. Note that motions across the overlapped resonances occur with high probabilities. Here, the fact that diffusion across resonances are faster than Arnold diffusion along the resonances, pointed out by Laskar [13], is clearly demonstrated as a global feature of Hamiltonian dynamical systems. 

Anomalous diffusion was first investigated in a one-dimensional chaotic map to describe enhanced diffusion in Josephson junctions [21], and it is observed in many systems both numerically [16,18,22-24] and experimentally [25], Anomalous diffusion is also observed in Hamiltonian dynamical systems. It is explained as due to power-type distribution functions [22,26,27] of trapping and untrapping times of the orbit in the self-similar hierarchy of cylindrical cantori [28]. 

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