Indeed, recent large-scale explorations of digital data have lifted the veil on many aspects of our artistic expressions that would remain forever hidden in smaller samples. And in such instances, methods of statistical physics and network science can be used to quantify and to better understand what it is that evokes that pleasant feeling, be it when reading a book or looking at a painting. But we all know or feel when something is beautiful to us personally.

The numerical results are supported by a spectral theory of convergence for weak phase modulations.īeauty is subjective, and as such it, of course, cannot be defined in absolute terms. The text presents numerous examples and tests of application for the method, covering multicomponent signals, observables of highly stable limit cycle oscillations and noisy phase dynamics. The work proposes a novel procedure called Iterative Hilbert Transform Embedding to obtain an optimal demodulation of signals. However, the Hilbert transform produces artificial variations of the reconstructed amplitude even in this case. For the ideal case of phase modulated signals, amplitude modulations vanish. The work presents a way to quantify how much the variations of signal amplitude spoil a phase demodulation procedure. Generally, observations of oscillations contain a small and generic variation of their amplitude. A popular approach for the construction of phases from signals is based on phase demodulation by means of the Hilbert transform. The numerical approach is finally tested on observation-based phase estimates of the Morris-Lecar neuron. A numerical study of high-order phase coupling applies a Fourier fit for the Stuart-Landau and for the van-der-Pol oscillator. A novel phase estimation scheme for direct observations of an oscillator dynamics is introduced based on numerics. Mathematical calculations consider the Stuart-Landau oscillator. The dissertation introduces and expands the knowledge about high-order phase coupling in networks of oscillators. One highly active field examines effects of non-local coupling among oscillators, which is thought to play a key role in networks with strong coupling. Notably, one can define phase models also for networks of oscillators. Experimentally, there exist well established approaches to extract phases from oscillator signals.

Theoretically these variables correspond to an integrable linearization of the generally non-linear system. Such variables are highly attractive for theoretical and experimental studies. Part of this theory considers reduced models of the oscillator dynamics by means of amplitudes and a phase variable. They emanate from non-linear systems in a heterogeneous environment and can be described by the theory of dynamical systems. Selfsustained oscillations are some of the most commonly observed phenomena in biological systems.

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