The Focus-Center-Limit Cycle Bifurcation in Discontinuous Planar Piecewise Linear Systems Without Sliding
Planar discontinuous piecewise linear systems with two linearity zones, one of them being of focus type, are considered. By using an adequate canonical form under certain hypotheses, the bifurcation of a limit cycle, when the focus changes its stability a
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Abstract Planar discontinuous piecewise linear systems with two linearity zones, one of them being of focus type, are considered. By using an adequate canonical form under certain hypotheses, the bifurcation of a limit cycle, when the focus changes its stability after becoming a linear center, is completely characterized. Analytic expressions for the amplitude, period and characteristic multiplier of the bifurcating limit cycle are provided. The studied bifurcation appears in real world applications, as shown with the analysis of an electronic Wien bridge oscillator without symmetry.
1 Introduction and Main Results Nowadays, the analysis of discontinuous piecewise-linear systems is an active field of research since certain modern devices are well modeled by this class of systems, see [5]. For the simplest situation however, as is the case of the aggregation of two planar linear systems, there are bifurcations that still require a thorough analysis. Recently, in [7] it has been proposed a canonical form for the case of planar discontinuous systems with two zones of linearity, to be denoted D2PWLS2 for short. In the quoted paper, there are shown some bifurcation results for the case when both linear dynamics are of focus type without visible tangencies, that is, there are no real equilibrium points in the interior of each half-plane. Here, by resorting to the canonical form given in [7], we consider a different situation when we have an equilibrium point of focus type in the interior of a half plane without specifying the linear dynamics type in the other half plane. Our goal is to describe qualitatively and quantitatively the possible bifurcation of limit cycles through the change of
E. Ponce () J. Ros E. Vela Departamento Matem´atica Aplicada II, E.T.S. Ingenier´ıa, 41092-Sevilla, Spain e-mail: [email protected]; [email protected]; [email protected] S. Ib´an˜ ez et al. (eds.), Progress and Challenges in Dynamical Systems, Springer Proceedings in Mathematics & Statistics 54, DOI 10.1007/978-3-642-38830-9 21, © Springer-Verlag Berlin Heidelberg 2013
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stability of such an equilibrium point. Thus, this work is a relevant generalization to discontinuous vector fields of the bifurcation studied in [6] for the continuous case. To begin with, we assume without loss of generality that the linearity regions in the phase plane are the left and right half-planes, S D f.x; y/ W x < 0g;
S C D f.x; y/ W x > 0g;
separated by the straight line ˙ D f.x; y/ W x D 0g. The systems to be studied become ( T F C .x/; F2C .x/ D AC x C bC ; if x 2 S C ; xP D 1 (1) T F1 .x/; F2 .x/ D A x C b ; if x 2 S ; where x D .x; y/T 2 R2 , AC D .aijC / and A D .aij / are 2 2 constant matrices, bC D .b1C ; b2C /T , b D .b1 ; b2 /T are constant vectors of R2 , and the definition of the vector field in ˙ is not relevant for our purposes. C We assume the generic condition a12 a12 > 0, which means that orbits can cross the discontinuity line in opposite directions, allowing the existence of period orbits that u
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