Number of Limit Cycles for Some Non-generic Classes of Piecewise Linear Differential Systems

Novaes, Douglas D.

doi:10.1007/978-3-319-55642-0_24

Cited by 2 publications

(3 citation statements)

References 10 publications

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“…This upper bound is decreased by two in the same cases in [23,24]. For this special class, the complete study is done in [28], where it is shown that the maximum number of limit cycles is two. Moreover, this upper bound is reached.…”

Section: Introductionmentioning

confidence: 97%

Limit cycles in planar piecewise linear differential systems with nonregular separation line

Cardin

Torregrosa

2016

Physica D: Nonlinear Phenomena

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Abstract. 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 α and 2π − α, respectively, for α ∈ (0, π). 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 α, we prove the existence of systems with four limit cycles up to fifth order and, for α = π/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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Section: Introductionmentioning

confidence: 97%

Limit cycles in planar piecewise linear differential systems with nonregular separation line

Cardin

Torregrosa

2016

Physica D: Nonlinear Phenomena

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“…Discontinuous piecewise linear systems with two regions separated by a straight line have received a lot of attention during the last years, see for instance [7,10,11,14,15,17,18] among other papers. In [10], the authors conjectured that piecewise linear systems with two regions separated by a straight line could have at most two crossing limit cycles.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Later on in [11], the authors provided numerical evidence on the existence of three crossing limit cycles, which was analytically proved in [17]. Sufficient condition on piecewise linear system implying the existence of at most 3 crossing limit cycles can be found in [7,15,18]. As far as we know, there are no examples of piecewise linear vector fields separated by a straight line with more than 3 crossing limit cycles.…”

Section: Introduction and Statement Of The Main Resultsmentioning

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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Number of Limit Cycles for Some Non-generic Classes of Piecewise Linear Differential Systems

Cited by 2 publications

References 10 publications

Limit cycles in planar piecewise linear differential systems with nonregular separation line

Limit cycles in planar piecewise linear differential systems with nonregular separation line

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

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