Abstract
In this paper, we study the limit cycles for m-piecewise discontinuous polynomial Liénard differential systems of degree n with m/2 straight lines passing through the origin whose slopes are \(\tan (\alpha + 2j\pi /m)\) for \(j = 0, 1, \ldots , m/2 -1\), and prove that for any positive even number m, if \(\sin ( m\alpha /2)\ne 0\), then there always exists such a system possessing at least \(\left[ \frac{1}{2}(n-\frac{m-2}{2}) \right] \) limit cycles. This result verifies a conjecture proposed by Llibre and Teixerira (Z Angew Math Phys 66:51–66, 2015).
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G. Dong is supported by the NSFC of China Grant 11626113. C. Liu is supported by the NSFC of China Grant 11371269.
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Dong, G., Liu, C. Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations. Z. Angew. Math. Phys. 68, 97 (2017). https://doi.org/10.1007/s00033-017-0844-2
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DOI: https://doi.org/10.1007/s00033-017-0844-2