Synchronization

Synchronization of an ensemble of oscillators is a phenomenon present in systems of different fields, ranging from social and physical to biological and technological
systems. The most successful approach to describe how synchrony emerges in these
systems is given by the Kuramoto model. This model as it stands, however, assumes
oscillators of fixed natural frequencies and a homogeneous all-to-all coupling strength.
The Kuramoto model has been analytically discussed to address the synchronization
phenomena of coupled oscillators in the thermodynamic limit (N --> ∞). However,
there needs to be a modi cation to address the inevitable in
uence of external fields
on the pattern of various real life synchronization phenomena which, in general; involves a finite number of oscillators. This research introduces a time dependent
coupling strength K(t) which is from the modulation of external elds in the form
of, for example, a periodic impulse, in the nite oscillators assembly. A sinusoidal
function with some arbitrary values of amplitude and frequency is added to the fixed
coupling strength as a perturbation of external elds. Temporal evolution of order
parameter r(t) and phase θ(t), both of which measure the degree of synchronization
of an assembly of oscillators simultaneously, are compared between uniform and time dependent cases. Graphical comparison are made using a 2 oscillator system, a building block of any finite oscillators case. Also, similar comparisons are performed for
a system of 32 oscillators which are chosen randomly as a representative of a nite
number of oscillators (2 < N < ∞). A temporal variation of the relative phase angle
θ(t) = θ2(t) - θ1(t) in 2 and 32 oscillators systems using uniform and time dependent
cases is also a part of this research. This work also introduces a time-dependent
coupling strength in the form of a step function. The main objective of using such
a function is to keep the synchronized behavior of the oscillators persistently. This
behavior can be achieved with the perception that occasional boosting with higher
coupling strength K(t) should be enough to sustain synchronous behavior of oscillators which, in general, are tuned with lower K(t).

This dissertation investigated the nature of pulse in the tempo fluctuation of music performance and how people entrain with these performed musical rhythms. In Experiment 1, one skilled pianist performed four compositions with natural tempo fluctuation. The changes in tempo showed long-range correlation and fractal (1/f) scaling for all four performances. To determine whether the finding of 1/f structure would generalize to other pianists, musical styles, and performance practices, fractal analyses were conducted on a large database of piano performances in Experiment 3. Analyses revealed signicant long-range serial correlations in 96% of the performances. Analysis showed that the degree of fractal structure depended on piece, suggesting that there is something in the composition's musical structure which causes pianists' tempo fluctuations to have a similar degree of fractal structure. Thus, musical tempo fluctuations exhibit long-range correlations and fractal scaling. To examine how people entrain to these temporal fluctuations, a series of behavioral experiments were conducted where subjects were asked to tap the pulse (beat) to temporally fluctuating stimuli. The stimuli for Experiment 2 were musical performances from Experiment 1, with mechanical versions serving as controls. Subjects entrained to all stimuli at two metrical levels, and predicted the tempo fluctuations observed in Experiment 1. Fractal analyses showed that the fractal structure of the stimuli was reected in the inter-tap intervals, suggesting a possible relationship between fractal tempo scaling, pulse perception, and entrainment. Experiments 4-7 investigated the extent to which people use long-range correlation and fractal scaling to predict tempo fluctuations in fluctuating rhythmic sequences.