Семинар "Динамические системы", 3 апреля

[Dmitry Todorov]

Название доклада: "Bifurcation structures in domain of robust chaos in piecewise smooth maps"
Докладчик: Виктор Аврутин (Universität Stuttgart)
Описание: 

The theory of nonlinear dynamical systems was initially developed
mainly for models with smooth system function. Since mid of 1990th
piecewise smooth models became a focus or research interest as they
are providing an adequate description for many systems both in the nature
and engineering sciences. Power converters, nowadays present in every
computer, mechanical systems with hard impacts and/or frictions,
models of financial markets, and many other practical applications
lead us to investigate such systems. In the last twenty years it has been
shown that these systems possess many phenomena which can not occur in
smooth systems. Typical examples are border collision bifurcations,
sliding, chattering, just to list a few. These phenomena are caused by
the presence of so-called switching manifolds in the state space and
their interactions with several invariant sets.

Chaotic attractors in piecewise smooth systems differ in many respects
from chaotic attractors in smooth systems. In particular, it is known
that they may be robust (persistent under parameter perturbation).
Accordingly, in the parameter space of a piecewise smooth system we
may observe open regions associated with chaotic attractors only,
without any periodic windows. However, it would be wrong to see such a 
region (called also chaotic domain) as homogeneous. By
contrast, in these regions we observe bifurcation structures following
some generic patterns. Discussing these patterns is the the overall
goal of this talk.

First of all we consider some properties of chaotic attractors in
piecewise smooth systems. As we shall see, in continuous piecewise
smooth systems multi-band chaotic attractors (that means, attractors
with more than one connected component) are necessarily cyclic,
whereas in discontinuous piecewise smooth systems they may also be
acyclic (necessary and sufficient conditions for that are known). As a
next step, we consider the connection between transformation of
chaotic attractors and homoclinic bifurcations of repelling cycles.
Therefore, the concept of critical homoclinic orbits is discussed.
Based on that, we revise several types of bifurcations of chaotic
attractors (merging, expansion and final bifurcations, also known as
crises or contact bifurcations).  As a final step, we present some
recently discovered bifurcation patterns (bandcount doubling,
bandcount adding and bandcount incrementing scenarios), their
organizing principles and conditions for their appearance.


Ссылка на pdf версию анонса:
http://chebyshev.spb.ru/userfiles/file/DynSysSeminar/03_04_14_avrutin.pdf

Доклад состоится
в четверг,  3го апреля в 15:15
на 14 линии В.О., д. 29Б,
аудитория 38 (предположительно)

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Best regards, Dmitry Todorov