On the number of limit cycles in general planar piecewise linear systems

By Songmei Huan, Xiao-Song Yang. 2012

201 citations recorded by the source corpus.

Much progress has been made in planar piecewise smooth dynamical systems. However there remain many important problems to be solved even in planar piecewise linear systems. In this paper, we investigate the number of limit cycles of planar piecewise linear systems with two linear regions sharing the same equilibrium. By studying the implicit Poincare map induced by the discontinuity boundary, some cases when there exist at most 2 limit cycles is completely investigated. Especially, based on these results we provide an example along with numerical simulations to illustrate the existence of 3 limit cycles thus have a negative answer to the conjecture by M. Han and W. Zhang [11](J. Differ.Equations 248 (2010) 2399-2416) that piecewise linear systems with only two regions have at most 2 limit cycles.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f020bacf32be9043b60e3c819788c998b06ef0d4

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In Geometry and Topology

Citations in the corpus

Piecewise linear differential systems with only centers can create limit cycles? (2017, background): “It is known that such systems can have three limit cycles, see [4,5,9,11,14,16].”
Limit cycles in planar piecewise linear differential systems with nonregular separation line (2016, background): “The question is: Which is the maximum number of isolated periodic orbits, also called limit cycles, that bifurcate perturbing a center? This problem for piecewise differential systems defined in two zones have been studied recently, among other papers, in [5, 7, 17, 18, 19, 22, 25, 26, 32].”
On the birth of limit cycles for non-smooth dynamical systems (2012, methodology): “However, by considering a specific family of discontinuous PWL differential systems with two linear zones sharing the equilibrium position, in [14] strong numerical evidence about the existence of three limit cycles was obtained, and a proof was provided by Llibre and Ponce [19].”
Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center (2018, background): “There are examples of such systems exhibiting three limit cycles, see [5, 6, 10, 12, 17, 20], and at this moment we do not know if discontinuous piecewise linear differential systems separated by a straight line can have more than three limit cycles.”
New lower bound for the Hilbert number in piecewise quadratic differential systems (2018, methodology): “An example with 3 limit cycles was firstly detected numerically in [17] by Huan and Yang.”
Piecewise linear perturbations of a linear center (2013, background): “Limit cycles of discontinuous piecewise linear differential systems defined on two half-planes separated by a straight line have been studied recently in [5, 16, 17, 18, 21, 22], among other papers.”
The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems (2017, background): “In the discontinuous ones we can have more than one limit cycle, either all crossing cycles or including one sliding cycle, and in fact, the determination of the number of limit cycle has been the subject of several recent papers, see [2, 3, 4, 10, 15, 16, 17, 20, 22, 23, 24].”
The discontinuous matching of two planar linear foci can have three nested crossing limit cycles (2014, background): “As mentioned in the introduction, this situation was first detected in a rather specific example appeared in [13].”
Bifurcation of limit cycles at infinity in piecewise polynomial systems (2018, background): “Further, Huan and Yang [21], and Freire et al. [22] respectively proved P (1) ≥ 3.”
On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line (2014, background): “Llibre and Ponce in [20] inspired in the numerical example of [13] proved that there are discontinuous piecewise linear differential systems with two pieces separated by a straight line having three limit cycles.”