On Hopf bifurcation in non-smooth planar systems

By Maoan Han, Weinian Zhang. 2010

290 citations recorded by the source corpus.

Abstract As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincare map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/73843a108ea6b5ae6a9cc108176d55f5d60e4e4e

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Citations in the corpus

Bifurcation of limit cycles in piecewise smooth systems via Melnikov function (2015, background): “Usually, we suppose that the center Lα is elementary (see [2] for the definition of elementary singular point for piecewise smooth systems on the plane).”
The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems (2017, unknown): “There are analogous results for piecewise smooth systems, for the case of continuous systems see for example [6, 7, 26, 27], and for the case of discontinuous systems see [1, 8, 11, 12, 14, 18].”
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 discontinuous matching of two planar linear foci can have three nested crossing limit cycles (2014, background): “The Hopf bifurcation in non-smooth systems was studied in [12], [3] and [19].”
Uniqueness of limit cycles for sewing planar piecewise linear systems (2015, background): “Recently, examples of piecewise linear systems with a sliding segment that present two and three limit cycles are provided, see among others [3, 13, 14, 15] and [5, 7, 19], respectively.”
On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line (2014, background): “Limit cycles of discontinuous piecewise linear differential systems with two pieces separated by a straight line have been studied by many authors, see for instance the articles [2, 8, 9, 11, 13, 14, 15, 20, 21, 24].”
THREE NESTED LIMIT CYCLES IN DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH TWO ZONES (2012, background): “Discontinuous piecewise linear systems with only two linearity regions separated by a straight line have been studied recently in [7, 8], among other papers.”
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].”
Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center (2018, background): “Limit cycles of discontinuous piecewise linear differential systems separated by a straight line have been studied by many authors, see for instance [2, 8, 9, 11, 13, 14, 21, 26] and the references therein.”