On the number of periodic solutions induced by a singular periodic solution (Q2703947)

Consider a differential equation NEWLINE\[NEWLINE z'=z^n+p(t,z),\tag{1}NEWLINE\]NEWLINE where \(p\) is a polynomial in \(z\) of degree \(n-1\) such that its coefficients are \(\omega\)-periodic in \(t\). The author defines a so-called transverse singular periodic solution with a single component (TSPS-1) for equation (1). Let \(g(s)\), \(s\in[-1,1]\), be a continuous family of equations (1) with the following properties: NEWLINENEWLINENEWLINE(i) \(g(0)\) has a TSPS-1 and no other singular periodic solutions, NEWLINENEWLINENEWLINE(ii) for \(s\neq 0\), equations \(g(s)\) have no singular periodic solutions. NEWLINENEWLINENEWLINEIt is shown that \(|N(1)-N(-1)|\leq 1\), where \(N(s)\) is the number of \(\omega\)-periodic solutions to the family of equations \(g(s)\).