56 research outputs found
Limit cycles from a monodromic infinity in planar piecewise linear systems
Planar piecewise linear systems with two linearity zones separated by a
straight line and with a periodic orbit at infinity are considered. By using
some changes of variables and parameters, a reduced canonical form with five
parameters is obtained. Instead of the usual Bendixson transformation to work
near infinity, a more direct approach is introduced by taking suitable
coordinates for the crossing points of the possible periodic orbits with the
separation straight line. The required computations to characterize the
stability and bifurcations of the periodic orbit at infinity are much easier.
It is shown that the Hopf bifurcation at infinity can have degeneracies of
co-dimension three and, in particular, up to three limit cycles can bifurcate
from the periodic orbit at infinity. This provides a new mechanism to explain
the claimed maximum number of limit cycles in this family of systems. The
centers at infinity classification together with the limit cycles bifurcating
from them are also analyzed.Comment: 24 pages, 5 figure
Bifurcation of Limit Cycles from a Periodic Annulus Formed by a Center and Two Saddles in Piecewise Linear Differential System with Three Zones
In this paper, we study the number of limit cycles that can bifurcate from a
periodic annulus in discontinuous planar piecewise linear Hamiltonian
differential system with three zones separated by two parallel straight lines,
such that the linear differential systems that define the piecewise one have a
center and two saddles. That is, the linear differential system in the region
between the two parallel lines (i.e. the central subsystem) has a center and
the others subsystems have saddles. We prove that if the central subsystem has
a real or a boundary center, then we have at least six limit cycles bifurcating
from the periodic annulus by linear perturbations, four passing through the
three zones and two passing through the two zones. Now, if the central
subsystem has a virtual center, then we have at least four limit cycles
bifurcating from the periodic annulus by linear perturbations, three passing
through the three zones and one passing through the two zones. For this, we
obtain a normal form for these piecewise Hamiltonian systems and study the
number of zeros of its Melnikov functions defined in two and three zonesComment: arXiv admin note: text overlap with arXiv:2109.1031
Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
In the present study we consider planar piecewise linear vector fields with
two zones separated by the straight line . Our goal is to study the
existence of simultaneous crossing and sliding limit cycles for such a class of
vector fields. First, we provide a canonical form for these systems assuming
that each linear system has center, a real one for and a virtual one for
, and such that the real center is a global center. Then, working with a
first order piecewise linear perturbation we obtain piecewise linear
differential systems with three crossing limit cycles. Second, we see that a
sliding cycle can be detected after a second order piecewise linear
perturbation. Finally, imposing the existence of a sliding limit cycle we prove
that only one adittional crossing limit cycle can appear. Furthermore, we also
characterize the stability of the higher amplitude limit cycle and of the
infinity. The main techniques used in our proofs are the Melnikov method, the
Extended Chebyshev systems with positive accuracy, and the Bendixson
transformation.Comment: 24 pages, 7 figure
Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones
Agraïments: FEDER-UNAB 10-4E-378. The second author is partially supported by a FAPESP-BRAZIL grant 2013/16492-0. The third authors is partially supported by a FAPESP-BRAZIL grant 2012/18780-0. The three authors are also supported by a CAPES CSF-PVE grant 88881.030454/ 2013-01 from the program CSF-PVE.We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0 we have a linear saddle with its equilibrium point living in x > 0, and in x 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential systems formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center
On the Limit Cycles for Continuous and Discontinuous Cubic Differential Systems
We study the number of limit cycles for the quadratic polynomial differential systems xË™=-y+x2, yË™=x+xy having an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones
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