10,710 research outputs found
On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line
In this paper we study the maximum number of limit cycles that can
exhibit a planar piecewise linear differential system formed by two pieces
separated by a straight line. More precisely, we prove that this maximum number
satisfies if one of the two linear differential systems has
its equilibrium point on the straight line of discontinuity
Shilnikov problem in Filippov dynamical systems
In this paper we introduce the concept of sliding Shilnikov orbits for D
Filippov systems. In short, such an orbit is a piecewise smooth closed curve,
composed by Filippov trajectories, which slides on the switching surface and
connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A
version of the Shilnikov's Theorem is provided for such systems. Particularly,
we show that sliding Shilnikov orbits occur in generic one-parameter families
of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit
there exist countably infinitely many sliding periodic orbits. Here, no
additional Shilnikov-like assumption is needed in order to get this last
result. In addition, we show the existence of sliding Shilnikov orbits in
discontinuous piecewise linear differential systems. As far as we know, the
examples of Fillippov systems provided in this paper are the first exhibiting
such a sliding phenomenon
Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems
Systems that are not smooth can undergo bifurcations that are forbidden in
smooth systems. We review some of the phenomena that can occur for
piecewise-smooth, continuous maps and flows when a fixed point or an
equilibrium collides with a surface on which the system is not smooth. Much of
our understanding of these cases relies on a reduction to piecewise linearity
near the border-collision. We also review a number of codimension-two
bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure
Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold
We study the family of piecewise linear differential systems in the plane
with two pieces separated by a cubic curve. Our main result is that 7 is a
lower bound for the Hilbert number of this family. In order to get our main
result, we develop the Melnikov functions for a class of nonsmooth differential
systems, which generalizes, up to order 2, some previous results in the
literature. Whereas the first order Melnikov function for the nonsmooth case
remains the same as for the smooth one (i.e. the first order averaged function)
the second order Melnikov function for the nonsmooth case is different from the
smooth one (i.e. the second order averaged function). We show that, in this
case, a new term depending on the jump of discontinuity and on the geometry of
the switching manifold is added to the second order averaged function
An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: the Averaging Method Revisited
This paper introduces an algorithmic approach to the analysis of bifurcation
of limit cycles from the centers of nonlinear continuous differential systems
via the averaging method. We develop three algorithms to implement the
averaging method. The first algorithm allows to transform the considered
differential systems to the normal formal of averaging. Here, we restricted the
unperturbed term of the normal form of averaging to be identically zero. The
second algorithm is used to derive the computational formulae of the averaged
functions at any order. The third algorithm is based on the first two
algorithms that determines the exact expressions of the averaged functions for
the considered differential systems. The proposed approach is implemented in
Maple and its effectiveness is shown by several examples. Moreover, we report
some incorrect results in published papers on the averaging method.Comment: Proc. 44th ISSAC, July 15--18, 2019, Beijing, Chin
On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in
In this paper we use the blow up method of Dumortier and Roussarie
\cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to
Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of
singularities of piecewise smooth dynamical systems
\cite{filippov1988differential} in . Using the regularization
method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the
power of our approach by considering the case of a fold line. We quickly
recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a
simple manner. Then, for the two-fold singularity, we show that the regularized
system only fully retains the features of the singular canards in the piecewise
smooth system in the cases when the sliding region does not include a full
sector of singular canards. In particular, we show that every locally unique
primary singular canard persists the regularizing perturbation. For the case of
a sector of primary singular canards, we show that the regularized system
contains a canard, provided a certain non-resonance condition holds. Finally,
we provide numerical evidence for the existence of secondary canards near
resonance.Comment: To appear in SIAM Journal of Applied Dynamical System
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