On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems

Gouveia, Márcio Ricardo Alves; Llibre, Jaume; Novaes, Douglas D.

doi:10.1016/j.amc.2015.09.022

Cited by 11 publications

(14 citation statements)

References 22 publications

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“…However when the planar vector field is discontinuous, the adaptation of the 16th Hilbert's problem on the maximum number of existing limit cycles, is an open problem. In the last years many authors have worked in this problem, trying to determine how many limit cycles can appear in planar systems separated by a straight line, see for instance [1][2][3][4]8,[10][11][12][13][14][16][17][18][21][22][23][24][25][26][27][28][29] .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

Esteban

Llibre

Valls

2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper we deal with discontinuous piecewise differential systems formed by two differential systems separated by a straight line when these two differential systems are linear centers (which always are isochronous) or quadratic isochronous centers. It is known that there is a unique family of linear isochronous centers and four families of quadratic isochronous centers. Combining these five types of isochronous centers we obtain fifteen classes of discontinuous piecewise differential systems.We provide upper bounds for the maximum number of limit cycles that these fifteen classes of discontinuous piecewise differential systems can exhibit, so we have solved the 16th Hilbert problem for such differential systems. Moreover in seven of the classes of these discontinuous piecewise differential systems the obtained upper bound on the maximum number of limit cycles is reached.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

Esteban

Llibre

Valls

2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Here, we explore the information on the maximum number of limit cycles that can be obtained by studying the periodic orbit at infinity and its possible bifurcations in the mentioned family of planar discontinuous piecewise linear systems with two zones separated by a straight line. Bifurcations from infinity for planar piecewise linear differential systems have been analyzed before in [23], and more recently in [19]. In [23] only continuous cases with two zones and three zones with symmetry were considered.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For the two-zones case, only one bifurcating limit cycle was detected, according to the well-known fact that there can be only one limit cycle in the class of continuous planar piecewise linear differential systems with two zones separated by a straight line, see [12]. In [19], the bifurcation from infinity is addressed for the case of discontinuous piecewise linear differential systems, by perturbing in a non-symmetric way the canonical continuous linear center ( ẋ, ẏ) = (−y, x), allowing for different linear perturbations in the half-planes y < 0 and y > 0. Again, only one bifurcating limit cycle was obtained.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Limit cycles from a monodromic infinity in planar piecewise linear systems

Freire¹,

Ponce²,

Torregrosa³

et al. 2020

Preprint

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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.

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“…The Bendixson transformation is a useful tool to analyze the stability of the infinity of planar vector fields. In what follows, following [9,16], we shall discuss this transformation. Consider the differential systems…”

Section: 2mentioning

confidence: 99%

“…The difference between M − 1 and M + 1 provides the first order Melnikov function M 1 (y 0 ) = M − 1 (y 0 ) − M + 1 (y 0 ) given in (9), and the simple zeros of M 1 (y 0 ) provide the crossing limit cycles of Z 1,ε (X).…”

Section: 3mentioning

confidence: 99%

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

et al. 2020

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In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. 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 y < 0 and a virtual one for y > 0, 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.2010 Mathematics Subject Classification. 34C05, 34C07, 37G15.

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On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems

Cited by 11 publications

References 22 publications

The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

Limit cycles from a monodromic infinity in planar piecewise linear systems

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

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