New insight into bifurcation and chaos
Models with non-smooth functions have gained significant momentum as they provide an adequate description for many systems, in both 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 now, systems with one switching manifold have mainly been investigated. The EU-funded project 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. They then moved on to bifurcation structures that involve multi-band chaotic attractors, identifying the general mechanism leading the number of bands. Another part of the project work was geared toward the behaviour of DC/AC converters in view of an unusual transition from the domain of stable fixed points to the chaotic dynamic. The former domain corresponds to the converter's desired mode of operation. 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 related to 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 carried out with regard to segregation models describing the entry/exit of two populations into/from 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.
Keywords
Bifurcation, chaos, bifurcation structures, multi-dimensional parameter spaces, switching manifolds
