Full text
| |
| Author(s): |
Ana Elisa Dellamatrice Barioni
Total Authors: 1
|
| Document type: | Master's Dissertation |
| Press: | Campinas, SP. |
| Institution: | Universidade Estadual de Campinas (UNICAMP). Instituto de Física Gleb Wataghin |
| Defense date: | 2022-08-26 |
| Examining board members: |
Marcus Aloizio Martinez de Aguiar;
José Antonio Brum;
Adilson Enio Motter
|
| Advisor: | Marcus Aloizio Martinez de Aguiar |
| Abstract | |
The dynamics of large systems of coupled oscillators is a subject of increasing importance with prominent applications in several areas such as physics and biology. Kuramoto’s original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. This model, where a set of oscillators move around a circle representing their phases, is a paradigm in this field, exhibiting a continuous transition between disordered and synchronous motion. Reinterpreting the oscillators as rotating unit vectors, the model was extended to higher dimensions by allowing vectors to move on the surface of D-dimensional spheres, with D=2 corresponding to the original model. It was shown that the transition to synchronous dynamics was discontinuous for odd D. Inspired by results in 2D, Ott et al proposed an ansatz for the density function describing the oscillators and derived equations for the ansatz parameters, effectively reducing the dynamics complexity. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this project we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. We take a different approach for the 3D system and construct an ansatz based on spherical harmonics decomposition of the distribution function. Our result differs from Ott’s work and leads to similar but simpler equations determining the dynamics of the order parameter. We derive the phase diagram of equilibrium solutions for several distributions of natural frequencies and find excellent agreement with numerical solutions for the full system dynamics. In the case of motion in a D-dimensional sphere the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz (AU) | |
| FAPESP's process: | 19/24068-0 - Generalized Kuramoto oscillators with external forces |
| Grantee: | Ana Elisa Dellamatrice Barioni |
| Support Opportunities: | Scholarships in Brazil - Master |