On the number of limit cycles in piecewise smooth generalized Abel equations with two asymmetric zones

Abstract

This paper devotes to the study of a kind of generalized Abel equation dxdθ=ap(θ)xp+aq(θ)xq,where p,q∈Z∖{1}, q−1p−1∉Z≤1, and ap, aq are piecewise trigonometrical polynomials of degree m with two zones 0≤θ<θ1 and θ1≤θ≤2π. We focus on the maximum number of positive and negative limit cycles (i.e., positive and negative isolated periodic solutions) that the equation can have, and the problem that how this maximum number, denoted by Hθ1(m), is affected by the location of the separation line θ=θ1. Then, by virtue of arbitrary higher order analysis using the theories of Melnikov functions and Chebyshev systems, we obtain lower bounds for Hθ1(m) that, Hθ1(m)≥2(3m+1) (resp. Hθ1(m)≥3m+1) when θ1∈0,π∪π,2π and either p or q is odd (resp. both p and q are even), and Hθ1(m)≥4m (resp. Hθ1(m)≥2m) when θ1=π and either p or q is odd (resp. both p and q are even). This result includes the estimates for not only the classical Abel equations but also some other equations from the real problems, such as the pendulum-like equations. In general, the asymmetry of the two zones increases the number of limit cycles in comparison with the case where the two zones are symmetric.