• caos
• chaos
• dynamical systems
• oscillatori a rilassamento
• oscillatori UJT
• relaxation oscillators
• sistemi dinamici
• transition to chaos
• transizione al caos
• UJT oscillators

The focus of this thesis consists in the modelling and in the study of the dynamical behaviour of a two-dimensional nonlinear relaxation oscillator represented by a system of two coupled nonlinear differential equations. In general, nonlinear dynamical models are designed to describe systems that cannot be accurately captured by linear models. Some of the peculiarities of such systems include the presence of periodic or quasiperiodic oscillations and can display, in particular conditions, complex and unpredictable behaviours. Nonlinear oscillator models can be found in a variety of disciplines including electronics, geology, chemistry and biology, where most oscillators are of the relaxation type, and in particular, in models of spiking neurons. In electronic circuits the presence of nonlinear active elements enables the emergence of self-sustained oscillations and in this context, the UJT (Uni Junction Transistor) can be employed to construct a simple relaxation oscillator where the junction acts as an electrically controlled switch that allows the current to flow only after a triggering potential point. In this circuit the UJT connects a capacitor and a load resistor, where the charge and the current are measured. The resulting waveforms present two distinct time scales corresponding to the charge and discharge of the capacitor and the impulsive current flowing through the resistor creating a two-stroke oscillator. Although such system was widely studied, at present we do not have a satisfactory theoretical model yet to describe its behaviour. Therefore, the search for a mathematical model capable of reproducing qualitatively and quantitatively the experimental data was one objective of this thesis. To this aim a new mathematical model of the UJT circuit, represented by two coupled nonlinear ordinary differential equations, was formulated. The corresponding parameters were determined by fitting the experimental data. The physical quantities defining the state of this two-dimensional dynamical system were the capacitor charge and the discharge current of the UJT component through the load resistor. Next, this model was studied analytically within the framework of dynamical systems and so, in the thesis, we focused on investigating the stationary states, their stability and the conditions for the existence of a limit cycle. Finally, as done experimentally, this dynamical system was perturbed to study the emergence of complex phenomena such as the phase locking, the quasiperiodicity and the chaotic behaviours. In addition, the synchronization properties of two coupled UJT oscillator were investigated through the theory of phase reduction of nonlinear oscillators. The work developed in this thesis is a continuation of previous works which focused on the UJT relaxation two-stroke oscillator and, with respect to them, a new and original model of the UJT circuit was proposed and its dynamical properties were studied analytically. This continuous model, although conceptually simple, can reproduce the UJT dynamical behaviour and the transition to chaos as empirically observed.