Synchronization of physical oscillators

Tese de mestrado, Física (Física da Matéria Condensada e Nanomateriais) Universidade de Lisboa, Faculdade de Ciências, 2018The phenomenon of synchronization can be found in several biological and non-biological systems, such as, for example, in physical pendulum clocks, metronomes, pacemaker cells, firefly interaction and planet orbits. In this work we present a study concerning the synchronization of two coupled pendulum clocks. We use techniques of Dynamical Systems, in particular, when we study the Andronov model for an oscillator with impulses. We study the cases for a frequency relation of 1 : 1. We analyze the amplitude of the limit cycles of each oscillator and find an asymptotic stable fixed point for the velocity of an isolated clock. In a study of the phase, we construct the functions that map the phase difference between two coupled clocks and analyze its evolution, for a frequency relation of 1 : 1. We do the same study for a frequency relation of 2 : 1, which is a a study made by the first time in this work. We find one stable fixed point and one unstable fixed point, both for frequency relations of 1 : 1 and 2 : 1. We conclude that the system of coupled Andronov clocks tend to synchronize in phase opposition for the case 1 : 1 and in generalized phase opposition for the case 2 : 1. Also, we make a construction by phase approximation for the first order of the linear expansion, for frequency relations of N : 1. We then expand in second order the maps for a frequency relation of 1 : 1. With this approach, the results coincide with the more complicated technique of studying the original model, this technique is more suitable for numerical studies. Under the conditions of our problem, we conjecture that we have a master-slave relation for a frequency relation of N : 1, for N > 1, based on the observations of the maps of the phase difference and on the analysis of the linear expansions. We construct the regions of synchronization in the parameter space – the Arnold tongues – for two coupled Andronov clocks with frequency relations of 1 : 1, 2 : 1 and 3 : 1. We also analyze what effect does the friction have in these synchronization regions, which, in the synchronization of two Andronov clocks, is a new observation. Finally, we study the amplitude again, but now we do not assume the occurrence of phase locking. The orbits of the unperturbed system of coupled clocks can be described in the R2 surface of an invariant torus in phase space. We introduce the a Kolmogorov-Arnold-Moser like theory to verify that under a small limited perturbation to the system, the structure of the torus surface remains stable, which is, again, a new approach for the Andronov model