Ph.D. received on: 16/4/1998

E-mail: gragnani@elet.polimi.it

Tutor: Prof. S. Rinaldi, Politecnico di Milano
 
 

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Complex hystereses in dynamical systems   ___________________________________________________________________________________________________________

Advisor:

Prof. S. Rinaldi, Politecnico di Milano

Referees:

Prof. S. Salsa, Politecnico di Milano
 

Summary:

A formal definition of hysteresis is proposed for finite-dimensional continuous-time dynamical systems depending upon a parameter p. From this definition a formal rule for classifying hystereses is derived. A system has an hysteresis if it has two different attractors A₁ and A₂ depending continuously upon p in two overlapping ranges: p > p₁ and p < p₂ with p₁ < p₂. The hysteresis is called simple if both attractors are equilibria for all p. Consistently, it is complex if A₁ or A₂ is a cycle, or a torus or a strange attractor for some value of p. Hystereses are characterized by two catastrophic bifurcations of the system concerning A₁ and A₂ at p₁ and p₂, respectively. Some non-catastrophic bifurcations can also be involved in complex hystereses. Thus, an hysteresis can be identified by two sequences of non-catastrophic bifurcations followed by a catastrophic one. Numerous examples of different types of hysteresis are presented. Such examples concern biological and social sciences and have been published in international journals.
 
 

 
MINORS

A non standard application of the singular perturbation method 
 

Advisor: S. Baldone, Politecnico di Milano

Models of the interaction between population, economy, and environment often contain nonlinear functional relationships and variables that vary at different speeds. These properties imply apparent unpredictabilities in system behavior. Using a simple deterministic model of demographic, economic and environmental interactions, the usefulness of singular perturbation theory and local bifurcation theory is illustrated. In particular it is shown how it is possible to obtain analytic expressions for: (1) the critical level of pollution flow above which environmental deterioration is expected, (2) the time it takes for going from the critical level of pollution flow to the beginning of a rapid environmental deterioration, and (3) the level of pollution flow at the time that rapid deterioration begins. Since the results are analytic, they make the outcomes of demographic, economic, and environmental interactions more predictable, and, therefore, potentially more manageable.

 
Analysis and interpretation of experimental complex dynamics in a batch reactor 

Advisor: Prof. M. Morbidelli, Politecnico di Milano

In the last decades, chemical reactions, in which complex dynamics are observed, have been extensively theoretically and experimentally analyzed. After the discover of the Belousov-Zhabotinskii reaction, many other reactions have shown a variety of dynamic behaviors like periodic oscillations and chaos. Investigations have followed two different approaches. On one hand, more complex chinetic models, describing quantitatively the observed phenomena, have been proposed. On the other hand, simple models, reproducing qualitatively the observed phenomena, have been developed. This work follows the second approach. Making reference to a series of laboratory experiments on a batch reactor, the dynamic behavior of the BZ reaction, characterized by recurrent changes between stationary and cyclic regimes, has been interpreted by a simplified dynamical model. In particular, the role of initial concentrations of the reactants has been studied.
 

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