In these notes, we provide a perspective on the theoretical framework built on the Kuramoto model, a paradigm in the study of synchronized behavior in coupled oscillator models. Following the original works of Kuramoto, we discuss in depth assumptions and analytical approaches in his theory and explain all the steps involved in the derivation of the main results. Starting from analyzing the phase dynamics of synchronized and desynchronized oscillators, we obtain the distribution of phases and the order parameter equation. We also show how a critical coupling strength, associated to a transition to collective oscillation, emerges in the theory. Another important result, not yet addressed in reviews, is the distribution of ‘coupling-modified frequencies’, i.e. the instantaneous frequencies averaged over an infinitely long time. We introduce the concept of a coupling-modified frequency and obtain the corresponding distribution in a different way to Kuramoto’s works. For the particular case of a Cauchy–Lorentz distribution of natural frequencies, an extensive set of analytical results are explicitly obtained. Finally, we compare theoretical predictions of the order parameter with numerical data, showing the strength of the theory in the description of stationary states and discussing the limitations deriving from sources of fluctuations.

Some properties of the annular billiard under the presence of weak dissipation are studied. We show, in a dissipative system, that the average energy of a particle acquires higher values than its average energy of the conservative case. The creation of attractors, associated with a chaotic dynamics in the conservative regime, both in appropriated regions of the phase space, constitute a generic mechanism to increase the average energy of dynamical systems.

Shearless transport barriers appear in confined plasmas due to non-monotonic radial profiles and cause localized reduction of transport even after they have been broken. In this paper we summarize our recent theoretical and experimental research on shearless transport barriers in plasmas confined in toroidal devices. In particular, we discuss shearless barriers in Lagrangian magnetic field line transport caused by non-monotonic safety factor profiles. We also discuss evidence of particle transport barriers found in the TCABR Tokamak (University of São Paulo) and the Texas Helimak (University of Texas at Austin) in biased discharges with non-monotonic plasma flows.

We identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics (RTS), which is based on the concept of Poincaré recurrences. The projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near-integrable system, the annular billiard. These microstructures interfere in the effective transport of a particle in the phase space, which can be observed through RTS. This technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea.