# Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations 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 $$\tan (\alpha + 2j\pi /m)$$ tan ( α + 2 j π / m ) for $$j = 0, 1, \ldots , m/2 -1$$ j = 0 , 1 , … , m / 2 - 1 , and prove that for any positive even number m, if $$\sin ( m\alpha /2)\ne 0$$ sin ( m α / 2 ) ≠ 0 , then there always exists such a system possessing at least $$\left[ \frac{1}{2}(n-\frac{m-2}{2}) \right]$$ 1 2 ( n - m - 2 2 ) limit cycles. This result verifies a conjecture proposed by Llibre and Teixerira (Z Angew Math Phys 66:51–66, 2015). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Zeitschrift für angewandte Mathematik und Physik Springer Journals

# Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

, Volume 68 (4) – Aug 7, 2017
8 pages

/lp/springer_journal/note-on-limit-cycles-for-m-piecewise-discontinuous-polynomial-li-nard-JzoGo2hIVo
Publisher
Springer International Publishing
Subject
Engineering; Theoretical and Applied Mechanics; Mathematical Methods in Physics
ISSN
0044-2275
eISSN
1420-9039
D.O.I.
10.1007/s00033-017-0844-2
Publisher site
See Article on Publisher Site

### 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 $$\tan (\alpha + 2j\pi /m)$$ tan ( α + 2 j π / m ) for $$j = 0, 1, \ldots , m/2 -1$$ j = 0 , 1 , … , m / 2 - 1 , and prove that for any positive even number m, if $$\sin ( m\alpha /2)\ne 0$$ sin ( m α / 2 ) ≠ 0 , then there always exists such a system possessing at least $$\left[ \frac{1}{2}(n-\frac{m-2}{2}) \right]$$ 1 2 ( n - m - 2 2 ) limit cycles. This result verifies a conjecture proposed by Llibre and Teixerira (Z Angew Math Phys 66:51–66, 2015).

### Journal

Zeitschrift für angewandte Mathematik und PhysikSpringer Journals

Published: Aug 7, 2017

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