93 research outputs found
Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can one say what should happen?
We consider a piecewise smooth system in the neighborhood of a co-dimension 2
discontinuity manifold . Within the class of Filippov solutions, if
is attractive, one should expect solution trajectories to slide on
. It is well known, however, that the classical Filippov
convexification methodology is ambiguous on . The situation is further
complicated by the possibility that, regardless of how sliding on is
taking place, during sliding motion a trajectory encounters so-called generic
first order exit points, where ceases to be attractive.
In this work, we attempt to understand what behavior one should expect of a
solution trajectory near when is attractive, what to expect
when ceases to be attractive (at least, at generic exit points), and
finally we also contrast and compare the behavior of some regularizations
proposed in the literature.
Through analysis and experiments we will confirm some known facts, and
provide some important insight: (i) when is attractive, a solution
trajectory indeed does remain near , viz. sliding on is an
appropriate idealization (of course, in general, one cannot predict which
sliding vector field should be selected); (ii) when loses attractivity
(at first order exit conditions), a typical solution trajectory leaves a
neighborhood of ; (iii) there is no obvious way to regularize the
system so that the regularized trajectory will remain near as long as
is attractive, and so that it will be leaving (a neighborhood of)
when looses attractivity.
We reach the above conclusions by considering exclusively the given piecewise
smooth system, without superimposing any assumption on what kind of dynamics
near (or sliding motion on ) should have been taking place.Comment: 19 figure
Qualitative Analysis of Polycycles in Filippov Systems
In this paper, we are concerned about the qualitative behaviour of planar
Filippov systems around some typical minimal sets, namely, polycycles. In the
smooth context, a polycycle is a simple closed curve composed by a collection
of singularities and regular orbits, inducing a first return map. Here, this
concept is extended to Filippov systems by allowing typical singularities lying
on the switching manifold. Our main goal consists in developing a method to
investigate the unfolding of polycycles in Filippov systems. In addition, we
applied this method to describe bifurcation diagrams of Filippov systems around
certain polycycles
Piecewise Smooth Dynamical Systems Theory: The Case of the Missing Boundary Equilibrium Bifurcations
Canards in stiction: on solutions of a friction oscillator by regularization
We study the solutions of a friction oscillator subject to stiction. This
discontinuous model is non-Filippov, and the concept of Filippov solution
cannot be used. Furthermore some Carath\'eodory solutions are unphysical.
Therefore we introduce the concept of stiction solutions: these are the
Carath\'eodory solutions that are physically relevant, i.e. the ones that
follow the stiction law. However, we find that some of the stiction solutions
are forward non-unique in subregions of the slip onset. We call these solutions
singular, in contrast to the regular stiction solutions that are forward
unique. In order to further the understanding of the non-unique dynamics, we
introduce a regularization of the model. This gives a singularly perturbed
problem that captures the main features of the original discontinuous problem.
We identify a repelling slow manifold that separates the forward slipping to
forward sticking solutions, leading to a high sensitivity to the initial
conditions. On this slow manifold we find canard trajectories, that have the
physical interpretation of delaying the slip onset. We show with numerics that
the regularized problem has a family of periodic orbits interacting with the
canards. We observe that this family has a saddle stability and that it
connects, in the rigid body limit, the two regular, slip-stick branches of the
discontinuous problem, that were otherwise disconnected.Comment: Submitted to: SIADS. 28 pages, 12 figure
Stochastic Perturbations of Periodic Orbits with Sliding
Vector fields that are discontinuous on codimension-one surfaces are known as
Filippov systems and can have attracting periodic orbits involving segments
that are contained on a discontinuity surface of the vector field. In this
paper we consider the addition of small noise to a general Filippov system and
study the resulting stochastic dynamics near such a periodic orbit. Since a
straight-forward asymptotic expansion in terms of the noise amplitude is not
possible due to the presence of discontinuity surfaces, in order to
quantitatively determine the basic statistical properties of the dynamics, we
treat different parts of the periodic orbit separately. Dynamics distant from
discontinuity surfaces is analyzed by the use of a series expansion of the
transitional probability density function. Stochastically perturbed sliding
motion is analyzed through stochastic averaging methods. The influence of noise
on points at which the periodic orbit escapes a discontinuity surface is
determined by zooming into the transition point. We combine the results to
quantitatively determine the effect of noise on the oscillation time for a
three-dimensional canonical model of relay control. For some parameter values
of this model, small noise induces a significantly large reduction in the
average oscillation time. By interpreting our results geometrically, we are
able to identify four features of the relay control system that contribute to
this phenomenon.Comment: 44 pages, 9 figures, submitted to: J Nonlin. Sc
Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles
This article describes the implementation of a novel method for detection and continuation of bifurcations in non- smooth complex dynamic systems -- The method is an alternative to existing ones for the follow-up of associated phe- nomena, precisely in the circumstances in which the traditional ones have limitations (simultaneous impact, Filippov and first derivative discontinuities and multiple discontinuous boundaries) -- The topology of cycles in non-smooth sys- tems is determined by a group of ordered segments and points of different regions and their boundaries -- In this article, we compare the limit cycles of non-smooth systems against the sequences of elements, in order to find patterns -- To achieve this goal, a method was used, which characterizes and records the elements comprising the cycles in the order that they appear during the integration process -- The characterization discriminates: a) types of points and segments; b) direction of sliding segments; and c) regions or discontinuity boundaries to which each element belongs -- When a change takes place in the value of a parameter of a system, our comparison method is an alternative to determine topo- logical changes and hence bifurcations and associated phenomena -- This comparison has been tested in systems with discontinuities of three types: 1) impact; 2) Filippov and 3) first derivative discontinuities -- By coding well-known cy- cles as sequences of elements, an initial comparison database was built -- Our comparison method offers a convenient approach for large systems with more than two regions and more than two sliding segment
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