MUDIBI

Multiple-Discontinuity Induced Bifurcations in Theory and Applications

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 Obiettivo del progetto (Objective)

'The theory or nonlinear dynamical systems was initially developed mainly for models with smooth system function. Since mid of 1900th non-smooth models became a focus or research interest as they provide an adequate description for many systems both in the nature and in engineering sciences. In the last twenty years it was shown that these systems possess many phenomena which can not occur in smooth systems. These phenomena are caused by the presence of so-called switching manifolds in the state space and their interactions with several invariant sets. However, until now mainly systems with one switching manifold were investigated. This represents an important intermediate step but is often not sufficient for explanation of the behaviour of many systems of practical interest. To understand possible behaviours of these systems and to achieve a desired behaviour via a suitable system design, it is necessary to understand the bifurcation structures occurring in multi-dimensional parameter spaces. For the bifurcation structures caused by interactions of invariant sets with one switching manifold, many valuable results were obtained in the last time. The proposed project addresses the following question: What are the basic principles organizing the bifurcation structures in low-dimensional maps with more than one switching manifold? The goal of the proposed project is to explain some generic bifurcation structures characteristic for these systems using a suitable extension of some novel investigation techniques developed for maps with one switching manifold with the concept of organizing centres in multi-dimensional parameter spaces. The practical relevance of the obtained results will be demonstrated by applications not only from the Engineering Science but mainly from the fields of Economics, Financial Modeling and Social Sciences.'

Introduzione (Teaser)

An EU-funded project shed further insight into the bifurcation structures occurring in multidimensional parameter spaces. Project work should have important implications in engineering, economics and social sciences.

Descrizione progetto (Article)

Models with non-smooth functions have gained significant momentum as they provide an adequate description for many systems, both in nature and engineering. The phenomena occurring in such systems are caused by the presence of switching manifolds in the state space and their interactions with several invariant sets. Until currently, mainly systems with one switching manifold have been investigated.

The EU-funded project http://www.econ.uniurb.it/MuDiBi_MarieCurie/ (MUDIBI) (Multiple-discontinuity induced bifurcations in theory and applications) addressed the basic principles organising the bifurcation structures in low-dimensional maps with more than one switching manifold. The project provided an explanation for some generic bifurcation structures characteristic of these systems. MUDIBI used suitable techniques developed for maps with one switching manifold and applied the concept of organising centres in multidimensional parameter spaces.

Scientists investigated three generic bifurcation structures and obtained analytical expressions for the periodicity region boundaries using the map replacement technique. Then, they moved on to bifurcation structures that involve multi-band chaotic attractors, identifying the general mechanism leading the number of bands.

Another part of project work was geared toward the behaviour of DC/AC converters considering an unusual transition from the domain of stable fixed points to the chaotic dynamic. The former domain corresponds to the desired mode of operation of the converter. Scientists identified irregular cascades of different border-collision bifurcations that have never been reported before. Remarkably, the results are also valid for models whose behaviour is affected by a high number of border points resulting from the mode of operation of DC/AC converters.

One of the most significant findings regarded the appearance of organising centres in 1D maps with an arbitrary number of discontinuities. Scientists found that their appearance can be predicted in the same way as in maps with only one discontinuity.

Significant work was also performed with regard to segregation models describing entry and exit of two populations into a system. Using numerical methods, the team investigated the border collision bifurcations generated by the upper limit of individuals that are allowed to enter the system.