37 research outputs found
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
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
The Filippov moments solution on the intersection of two and three manifolds
In this thesis, we study the Filippov moments solution for differential equations with discontinuous right-hand side. In particular, our aim is to define a suitable Filippov sliding vector field on a co-dimension manifold , intersection of two co-dimension manifolds with linearly independent normals, and then to study the dynamics provided by this selection. More specifically, we devote Chapter 1 to motivate our interest in this subject, presenting several problems from control theory, non-smooth dynamics, vehicle motion and neural networks. We then introduce the co-dimension case and basic notations, from which we set up, in the most general context, our specific problem. In Chapter 2 we propose and compare several approaches in selecting a Filippov sliding vector field for the particular case of nodally attractive: amongst these proposals, in Chapter 3 we focus on what we called \emph{moments solution}, that is the main and novel mathematical object presented and studied in this thesis. There, we extend the validity of the moments solution to attractive under general sliding conditions, proving interesting results about the smoothness of the Filippov sliding vector field on , tangential exit at first-order exit points, uniqueness at potential exit points among all other admissible solutions. In Chapter 4 we propose a completely new and different perspective from which one can look at the problem: we study minimum variation solutions for Filippov sliding vector fields in , taking advantage of the relatively easy form of the Euler-Lagrange equation provided by the analysis, and of the orbital equivalence that we have in the eventuality does not have any equilibrium points on it; we further remove this assumption and extend our results. In Chapter 5 examples and numerical implementations are given, with which we corroborate our theoretical results and show that selecting a Filippov sliding vector field on without the required properties of smoothness and exit at first-order exit points ends up dynamics that make no sense, developing undesirable singularities. Finally, Chapter 6 presents an extension of the moments method to co-dimension and higher: this is the first result which provides a unique admissible solution for this problem.Ph.D