On the number of limit cycles in piecewise smooth generalized Abel equations with many separation lines

Abstract

This paper investigates generalized Abel equations of the form $dx/d\theta =A\left(\theta \right)x^p+B\left(\theta \right)x^q$, where $p$, $q\in Z_{\ge 2}$, $p\ne q$, and $A\left(\theta \right)$ and $B\left(\theta \right)$ are piecewise trigonometrical polynomials of degree $m$ with $n-1\in N^{+}$ separation lines $0<{\theta }_1<{\theta }_2<\cdots <{\theta }_{n-1}<2\pi$. The main objective is to obtain the maximum number of non-zero limit cycles (i.e., non-zero isolated periodic solutions) that the equation can have, denoted by $H_{{\theta }_1,{\theta }_2,\dots ,{\theta }_{n-1}}\left(m\right)$, and to analyze how the number and location of separation lines ${\left\{{\theta }_i\right\}}_{i=1}^{n-1}$ affect $H_{{\theta }_1,{\theta }_2,\dots ,{\theta }_{n-1}}\left(m\right)$. By using the theories of Melnikov functions and ECT-systems, we obtain lower bounds for $H_{{\theta }_1,{\theta }_2,\dots ,{\theta }_{n-1}}\left(m\right)$. Our result extend those of Huang et al. who studied the special case of $n=2$, and reveal that the lower bounds decrease in the presence of pairs of symmetrical separation lines.