Three crossing limit cycles in planar piecewise linear systems with saddle-focus type

By Liping Li. 2014

55 citations recorded by the source corpus.

This paper presents an analysis on the appearance of limit cycles in planar Filippov system with two linear subsystems separated by a straight line. Under the restriction that the orbits with points in the sliding and escaping regions are not considered, we provide firstly a topologically equivalent canonical form of saddle-focus dynamic with five parameters by using some convenient transformations of variables and parameters. Then, based on a very available fourth-order series expansion of the return map near an invisible parabolic type tangency point, we show that three crossing limit cycles surrounding the sliding set can be bifurcated from generic codimensionthree singularities of planar discontinuous saddle-focus system. Our work improves and extends some existing results of other researchers.

Subfields: Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b75be9d90551a716203fadc14b9e29be765f7bac

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In Astronomy and Astrophysics

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 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.”
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].”