Degenerate Hopf bifurcations in discontinuous planar systems. (English) Zbl 0973.34033
The authors study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities. The problem of stability is treated in generic cases by computing some kind of Lyapunov constants. In this paper, the authors use the so-called \((R, \theta, p, q)\)- generalized polar coordinates and obtain some kind of Lyapunov constants in the discontinuous case. These constants can generate limit cycles for some concrete examples.
Reviewer: Jiancheng Zou (Beijing)
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D08 | Characteristic and Lyapunov exponents of ordinary differential equations |
| 34A36 | Discontinuous ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
References:
| [1] | Alwash, M. A.M.; Lloyd, N. G., Nonautonomous equations related to polynomial two-dimensional systems, Proc. Roy. Soc. Edinburgh Sect. A, 105, 129-152 (1987) · Zbl 0618.34026 |
| [2] | Andronov, A. A.; Leontovich, E. A.; Gordon, I. I.; Maĭer, A. G., Theory of Bifurcations of Dynamic Systems on a Plane (1973), Halsted: Halsted New York/Toronto · Zbl 0282.34022 |
| [3] | Andronov, A. A.; Vitt, A. A.; Khaĭkin, S.È., Theory of Oscillators (1996), Pergamon: Pergamon Oxford · Zbl 0188.56304 |
| [4] | Bautin, N. N., On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Amer. Math. Soc. Transl, 100, 397-413 (1954) |
| [5] | Blows, T. R.; Lloyd, N. G., The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 98, 215-239 (1984) · Zbl 0603.34020 |
| [6] | Broer, H. W.; Dumortier, F.; van Strien, S. J.; Takens, F., Structures in Dynamics (1991), North-Holland: North-Holland Amsterdam · Zbl 0746.58002 |
| [7] | Cima, A.; Gasull, A.; Mañosa, V.; Mañosa, F., Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math, 27, 471-501 (1997) · Zbl 0911.34025 |
| [8] | Coll, B.; Gasull, A.; Prohens, R., Differential equations defined by the sum of two quasi-homogeneous vector fields, Canad. J. Math, 49, 212-231 (1997) · Zbl 0990.34030 |
| [9] | Farr, W. W.; Li, C.; Labouriau, I. S.; Langford, W. F., Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem, SIAM J. Math. Anal, 20, 13-30 (1989) · Zbl 0682.58035 |
| [10] | Filippov, A. F., Differential Equations with Discontinuous Righthand Sides (1988), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0664.34001 |
| [11] | Gasull, A.; Guillamon, A.; Mañosa, V., An explicit expression of the first Liapunov and period constants with applications, J. Math. Anal. Appl, 211, 190-212 (1997) · Zbl 0882.34040 |
| [12] | Gasull, A.; Prohens, R., Effective computation of the first Lyapunov quantities for a planar differential equation, Appl. Math. (Warsaw), 24, 243-250 (1997) · Zbl 0882.34038 |
| [13] | Göbber, F.; Williamowski, K.-D., Ljapunov approach to multiple Hopf bifurcation, J. Math. Anal. Appl, 71, 333-350 (1979) · Zbl 0444.34040 |
| [14] | Hassard, B.; Wan, Y. H., Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl, 63, 297-312 (1978) · Zbl 0435.34034 |
| [15] | Liapunov, A. M., Stability of motion, Mathematics in Science and Engineering (1966), Academic Press: Academic Press New York · Zbl 0161.06303 |
| [16] | World Wide Web pages of Analysis of Non-Smooth Dynamical System, www.mi.uni-koeln.de/mi/Forschung/Kuepper/Forschung1.htm; World Wide Web pages of Analysis of Non-Smooth Dynamical System, www.mi.uni-koeln.de/mi/Forschung/Kuepper/Forschung1.htm |
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