Z. Angew. Math. Phys. (2017) 68:97
2017 Springer International Publishing AG
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Note on limit cycles for m-piecewise discontinuous polynomial Li´enard diﬀerential
and Changjian Liu
Abstract. In this paper, we study the limit cycles for m-piecewise discontinuous polynomial Li´enard diﬀerential systems of
degree n with m/2 straight lines passing through the origin whose slopes are tan(α +2jπ/m)forj =0, 1,...,m/2 − 1,
and prove that for any positive even number m,ifsin(mα/2) = 0, then there always exists such a system possessing at
limit cycles. This result veriﬁes a conjecture proposed by Llibre and Teixerira (Z Angew Math Phys
Mathematics Subject Classiﬁcation. 34C29, 34C25.
Keywords. Limit cycle, Piecewise diﬀerential system, Li´enard diﬀerential system, Averaging method.
1. Introduction and statement of the main results
In recent years, the non-smooth diﬀerential systems have been studied extensively. They appear and
play an intrinsic role in a wide range of science areas, not only in Mathematics, but also in Physics and
Engineering, for instance, in control systems, mechanical systems, nonlinear oscillations, and particular
electrical circuits. For more details on them, one can see [1,2,7] and the references therein. No matter
what in the theories or in the applications of non-smooth diﬀerential systems, the detection of limit cycles
is of fundamental importance.
A kind of typical non-smooth diﬀerential systems is the following so-called m-piecewise discontinuous
Li´enard polynomial diﬀerential systems of degree n,
˙x = y +sgn(g
(x, y))F (x),
˙y = −x,
where F (x) is a polynomial of degree n and the zero set of the function sgn(g
(x, y)) with positive even
number m is the union of m/2 diﬀerent straight lines passing through the origin of coordinates dividing
the plane into sectors of angle 2π/m. Here, sgn(z) denotes the sign function, i.e.,
The above systems in some sense generalize the class of Li´enard diﬀerential systems to the non-smooth
diﬀerential systems; this is just the reason for the name.
G. Dong is supported by the NSFC of China Grant 11626113. C. Liu is supported by the NSFC of China Grant