Ευστάθεια και χάος χαμιλτώνιων συστημάτων πολλών βαθμών ελευθερίας

Abstract

The main part of the thesis begins with Chapter 3, where new research results are presented which concern the regular and chaotic dynamics of Hamiltonian systems of few degrees of freedom. Results are described on the behavior of indices distinguishing organized from chaotic motion in these systems and a comparison is made with corresponding results in the international literature. Then, new findings are reported on the theory and application of the method of the Generalized Alignment Index GALI, which is one of the most basic discoveries of the thesis in nonintegrable Hamiltonian systems of 2 and 3 degrees of freedom. Chapter 5 deals with the presentation of original research results in Hamiltonian systems of many degrees of freedom. Here new methods are introduced for the study of regions of regular and chaotic behavior of multi degree of freedom systems with the primary aim of understanding the behavior of these systems in the thermodynamic limit to give an answer to the crucial question of whether the laws of Statistical mechanics hold in the case of multi dimensional Hamiltonian systems. The author studies how chaotic regions increase in size around unstable Simple Periodic Orbits (SPOs) in phase space, beyond a critical value of the energy, while the transition from limited to widespread chaos is indicated by the fact that in regions of different unstable SPOs the corresponding Lyapunov spectra converge to the same exponential – like function. Computing then the sum of the positive Lyapunov exponents, which corresponds to the so called Kolmogorov – Sinai entropy, it is shown that the systems that are studied in this thesis the KS entropy increases linearly as a function of the number of degrees of freedom N, thus confirming that it is an extensive quantity of Statistical Mechanics. Finally, the new method of the Linear Dependence Index (LDI) is introduced for distinguishing between regular and chaotic orbits and its advantages are described when compared with the methods of Chapters 3 and 4. It is worth mentioning also that many of the results of this thesis can be applied to the study of the dynamics of symplectic mappings, for which Mr. Antonopoulos developed a new method which combines his techniques with those of Evolutionary Algorithms, for determining the dynamical aperture radius for the stability of symplectic maps which describe the dynamics of high energy particle accelerators.