Center-focus problem for discontinuous planar differential equations. (English) Zbl 1077.34031

The authors study the following class of discontinuous planar systems of ordinary differential equations \[ (\dot{x},\dot{y})= \begin{cases}(-y+P^{+}(x,y),x+Q^{+}(x,y)) & \text{if }y\geq0,\\ (-y+P^{-}(x,y),x+Q^{-}(x,y)) & \text{if }y\leq0, \end{cases}\tag{1} \] where \(P^{+},P^{-},Q^{+},Q^{-}\) are analytic functions starting at least with second order terms. They are concerned with: the center-focus problem (whether the origin of (1) is either a center, an attractor or a repeller); the problem of determining the maximal number of (small amplitude) limit cycles which bifurcate from the origin under the variation of the parameters inside this class of systems. Such problems are solvable by the method of Lyapunov numbers. The main contribution of the article is a new method for determination of the Lyapunov numbers. Moreover, the proposed method is easy to be implemented in a computer-algebraic system. The article contains some nice applications of the new method to quadratic and Kukles systems.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34A36 Discontinuous ordinary differential equations
Full Text: DOI


[1] Andronov A. A., Theory of Bifurcations of Dynamic Systems on a Plane (1973)
[2] Andronov A. A., Theory of Oscillators (1987) · Zbl 0615.76119
[3] Bautin N. N., Amer. Math. Soc. Transl. 100
[4] Cartan E., Differential Forms (1970)
[5] DOI: 10.1142/S0218127499001231 · Zbl 1089.34512
[6] Coll B., Discr. Contin. Dyn. Syst. 6 pp 609–
[7] DOI: 10.1006/jmaa.2000.7188 · Zbl 0973.34033
[8] DOI: 10.1007/978-94-015-7793-9
[9] Françoise J. P., Ergod. Th. Dyn. Syst. 16 pp 87–
[10] DOI: 10.5565/PUBLMAT_41197_07 · Zbl 0892.34023
[11] DOI: 10.1006/jdeq.1998.3437 · Zbl 0943.34021
[12] DOI: 10.1216/rmjm/1021249441 · Zbl 1041.34016
[13] DOI: 10.1112/S0024610798006486 · Zbl 0922.34037
[14] DOI: 10.1006/jdeq.1998.3549 · Zbl 0926.34033
[15] DOI: 10.1007/BFb0103843
[16] Lunkevič V. A., Diff. Eqs. 4 pp 837–
[17] Lunkevič V. A., Diff. Eqs. 18 pp 563–
[18] Pleškan I. I., Diff. Eqs. 9 pp 1396–
[19] DOI: 10.1007/978-3-0348-8798-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.