Limit cycles of Abel equations of the first kind
Consider the scalar differential equation NEWLINE\[NEWLINE{dx\over dt}= \sum^m_{i=0} a_i(t) x^{n_i},\tag{\(*\)}NEWLINE\]NEWLINE where all \(a_i\) are \(T\)-periodic analytic functions, and \(1\leq n_i\leq n\). For any (given) polynomial \(Q(x)= x^{n_0}- \sum^m_{i=1} \alpha_i x^{n_i}\), \((*)\) can be written in the form NEWLINE\[NEWLINE{dx\over dt}= a_0(t) Q(x)+ R(t,x).\tag{\(**\)}NEWLINE\]NEWLINE Let \(W\) be the Wronskian of \(Q\) and \(R\), that is NEWLINE\[NEWLINEW(t,x)= Q(x) R_x(t,x)- R(t,x) Q'(x).NEWLINE\]NEWLINE The authors prove that if \(V\) is a connected component of NEWLINE\[NEWLINEU:= \{(t,x)\in \mathbb{R}^2: W(t,x) Q(x)\neq 0\},NEWLINE\]NEWLINE then there is at most one closed orbit of \((**)\) in \(V\). Under additional conditions which imply that closed orbits are either contained in \(U\) or disjoint with \(U\), it is proved that the maximum number of connected components of \(U\) is \(3n-1\) which implies an upper bound for the closed orbits of \((**)\).
scientific article; zbMATH DE number 6363023
scientific article