• 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.