136 research outputs found
On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems
Agraïments: The first author is partially supported by a PROCAD-CAPES grant 88881.068 462/2014-01 and by a FAPESP grant 2013/13344-0.Agraïments: MINECO/FEDER grant UNAB13-4E-1604. The third author is partially supported by a FAPESP grant 2012/10231-7. The three authors are also supported by a CAPES CSF-PVE grant 88881.030454/2013-01 from the program CSF-PVE.In this paper we consider the linear differential center (x',y')=(-y,x) class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. Using the Bendixson transformation we provide sufficient conditions to ensure the existence of a crossing limit cycle coming purely from the infinity. We also study the displacement function for a class of discontinuous piecewise smooth differential system
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 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
Cyclicity Near Infinity in Piecewise Linear Vector Fields Having a Nonregular Switching Line
Altres ajuts: acords transformatius de la UABIn this paper we recover the best lower bound for the number of limit cycles in the planar piecewise linear class when one vector field is defined in the first quadrant and a second one in the others. In this class and considering a degenerated Hopf bifurcation near families of centers we obtain again at least five limit cycles but now from infinity, which is of monodromic type, and with simpler computations. The proof uses a partial classification of the center problem when both systems are of center type
Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line
Acord transformatiu CRUE-CSICWe solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude
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
Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0
In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center x. = −y((x + y)/2), y. = x((x + y)/2) with m ≥ 0 under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree n with the discontinuity set {(x, y) ∈ R : xy = 0}. Using the averaging theory up to any order N, we give upper bounds for the maximum number of limit cycles in the function of m, n, N. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number
The local cyclicity problem : Melnikov method using Lyapunov constants
In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so M(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65
Limit cycles in planar piecewise linear differential systems with nonregular separation line
Agraïments: The first author is supported by FAPESP grant number 2013/24541-0 and CAPES grant number 88881.030454/2013-01 Program CSF-PVE and UNAB13-4E-1604.In this paper we deal with lanar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles and 2 - respectively, for (0,). We study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using Melnikov functions. These limit cycles appear studying higher order piecewise linear perturbations of a linear center. It is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five. Moreover, for these values of we prove the existence of systems with four limit cycles up to fifth order and, for =/2, we provide an explicit example with five up to sixth order. In general, the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line
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