• caos
• chaos
• dynamical systems
• nonlinear oscillators
• oscillatori a rilassamento
• oscillatori nonlineari
• oscillatori UJT
• relaxation oscillators
• sistemi dinamici
• UJT oscillators

Design and implementation of nonlinear components in electronic devices has been and it is still a research area of remarkable interest and the corresponding applications started a new unprecedented era of technological capabilities and data analysis that profoundly changed and is still changing our societies. A branch of this sector is represented by the research interest in nonlinear electronic oscillators, whose physical features are useful in the development of a whole series of electronic components, such as inverters, switching power supplies, dual-slope ADCs and function generators.
Regarding nonlinear oscillators, it is of crucial interest to understand if, once they are coupled to other parts of a circuit or are modulated by a signal they exhibit, within a certain range of values for some control parameters, chaotic dynamics characterized by the presence of sensitivity to initial conditions, and a substantial unpredictability on the time evolution of the physical quantities involved.
The theoretical part of the work is therefore to describe the system of interest by developing a model of differential equations and to analyze its characteristics in the framework of the theory of dynamical systems, so as to be able to determine the conditions for which such a chaotic regime is present or absent in the system.
The oscillator we are going to investigate is part of the two-stroke relaxation class, characterized by a nonlinear dynamics with two distinct phases, a slower one in which a capacitive component is typically loaded, and a faster one, in which there is a rapid discharge through the nonlinear component of the circuit. This class of oscillators is in contrast to that of four-strokes, in which the oscillatory dynamics are divided into four phases, two of which are slow and two are fast.
In our case, the nonlinear component we are going to model is a UJT (Uni Junction Transistor), an asymmetric p-n junction attached to three different terminals. The UJT has a peculiar voltage-current characteristic, with a negative slope in its central part, in this way the object can act as an insulator or can allow a sudden passage of current, depending on the working point in which it is found. The UJT component is therefore inserted into a circuit where there are external voltage suppliers, a capacitor and various resistances that allow its current-voltage characteristic to be adjusted.
The topic of this thesis consists in the study of a model of differential equations for the description of the above mentioned two-dimensional two-stroke relaxation oscillator circuit, capable of generating a chaotic dynamic once it is coupled to an external perturbation.
Properties of the two-dimensional model will be initially studied, which can be traced back to the dynamical features of the single decoupled oscillator in absence of any perturbation. We will see forwhich range of parameter values there is a stable oscillation, how stable it is by assessing the values of its corresponding Floquet multipliers and how they vary in the presence of bistability.
We will then proceed to the study of the model in the presence of external perturbations. In particular, when the oscillator is coupled to an external periodic forcing or when two identical relaxation oscillators are coupled. In both cases, to characterize the corresponding dynamical behaviours, we will proceed to the calculation of bifurcation diagrams for the maxima of the oscillations in the circuit under variation of the external coupling, combined with the calculation of Lyapunov exponents. This procedure will allow us to identify and quantify the presence of chaos in the dynamics of the system.
Finally, we will perform a comparison with the experimental data obtained at the laboratories of INO- CNR in Florence.