On the number of limit cycles in piecewise smooth generalized Abel equations with many separation lines (Q6608373)

The central object of this paper are generalized Abel ordinary differential equations \N\[\N\dot x=A(t)x^p+B(t)x^q, \N\]\Nwith natural numbers \(p,q>1\) satisfying \(q\neq p\), where \(A,B\) denote piecewise trigonometric polynomials of degree \(m\) having the \(n-1\) separation lines \(0<\theta_1<\ldots<\theta_{n-1}<2\pi\). \N\NThe authors are interested in the maximum number \(H_{\theta_1,\ldots,\theta_{n-1}}(m)\) of nonzero limit circles (that is, nonzero isolated periodic solutions) that such Abel equations can have, as well as how the number and the location of the separation lines \(\{\theta_i\}_{i=1}^{n-1}\) affects \(H_{\theta_1,\ldots,\theta_{n-1}}(m)\). Using the Wronskian and the method of Melnikov functions, lower bounds for \(H_{\theta_1,\ldots,\theta_{n-1}}(m)\) are obtained. These results extend earlier contribution addressing the special case \(n=2\). They furthermore reveal that the lower bounds decrease in the presence of pairs of symmetrical separation lines.