Computation of chaoticity indices in many degree of freedom systems

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2009 (EN)
Υπολογισμός δεικτών χαοτικότητας σε συστήματα πολλών βαθμών ελευθερίας
Computation of chaoticity indices in many degree of freedom systems

Τζιότζιος, Θωμάς Αθανασίου

Στη διπλωματική εργασία αυτή, μελετάμε τη συμπεριφορά των εκθετών Lyapunov και των δεικτών χαοτικότητας GALIk, FLI σε σύστημα συζευγμένων συμπλεκτικών απεικονίσεων Suris και σε Χαμιλτονιανό σύστημα αλυσίδας Klein-Gordon συζευγμένων ταλαντωτών. Επαληθεύουμε τις θεωρητικά αναμενόμενες θέσεις, όπου για κανονικές τροχιές όλοι οι εκθέτες Lyapunov είναι μηδενικοί, οι δείκτες GALIk φθίνουν με νόμο δύναμης (power law) εξαρτώμενο από τη διάσταση του χώρου φάσεων του συστήματος, από τη διάσταση του τόρου πάνω στον οποίο γίνεται η κίνηση και από το πλήθος των διανυσμάτων απόκλισης τα οποία είναι αρχικά εφαπτόμενα πάνω στον τόρο.
In the present thesis, we study the behaviour of the Lyapunov Exponents and the chaoticity indices GALIk, FLI in a system of coupled symplectic Suris maps and in a Hamiltonian Klein-Gordon chain. We verify the theoretically expected behaviour, since for regular orbits every Lyapunov exponent is zero, GALIk decay following power law dependend to the dimension of the phase space, the dimension of the torus hosting the regular motion and the num- ber of deviation vectors being initially tangent to the torus. Hence GALIk can indicate the dimension of the torus, so we have managed to detect low dimensional tori. FLI follows the behaviour FLI ∝ logt for every regular motion. Chaotic motion was accompanied by the existence of positive Lya- punov exponents, while GALIk decay following exponential rate, with the only exception of GALI2 for the case where the two maximal Lyapunov ex- ponents are equal, hence GALI2 remains constant. This was theoretically expected. However, this was a case where FLI increased following the rate of FLI ∝ logt, which indicates regular motion. Nevertheless, FLI followed the expected law of FLI ∝ t for every other chaotic orbit. Having observed the different behaviour followed by GALIk, FLI for the cased of chaotic or regular motion, we can use them to construct stability maps, in order to visualise the dynamics of the multidimensional systems, where phase space cannot be plotted. It can be seen that adding more and more oscillators drives the stability maps to an almost unchanging qualita- tively form, hence the approximation of the N → ∞ oscillator chain by a 7 system of N ≈ 11 coupled symplectic maps or a Hamiltonian Klein-Gordon chain of N ≈ 12 oscillators is a reasonable step. Increasing the number of phase space dimensions leads to more time (iterations for maps or time u- nits for Hamiltonian systems) needed for the stability maps to give a clearer image. Also, due to the increasing number of equations, the computation of every step is slower. Hence it would be extremely time consuming to study a system of many degrees of freedom. Consequently, the qualitative approx- imation of a large chain by a much smaller one is a quite useful property. In the system of symplectic maps, we have shown that while the number of oscillators N increases, the regular orbits turn into chaotic, but localised around the central oscillator, orbits with very small Lyapunov exponents. Hence, in many dimension systems, the stability maps provide us with in- formation about the localisation or not of the motion. We have also shown computationally, with the use of Lyapunov exponents, the chaotic diffusion of a chaotic orbit that initiates from the libration area and results in the chaotic layer around the separatrix after some time. Moreover, we have used stability maps as a tool for the detection of Discrete Breather and its stability.

Postgraduate Thesis / Μεταπτυχιακή Εργασία

Hamiltonian systems
Χαμιλτονιανά συστήματα
Symplectic maps
Συμπλεκτικές απεικονίσεις
Chaoticity indices
Δείκτες χαοτικότητας

Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (EL)
Aristotle University of Thessaloniki (EN)


Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης, Σχολή Θετικών Επιστημών, Τμήμα Φυσικής

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