Separatrix cycles and mutiple limit cycles in a class of quadratic systems
The three parameter family \(C_{ll}\) of quadratic systems given by \(\dot x= -y+ \delta x+ \ell x^ 2+ m xy\), \(\dot y= x+ x^ 2\) defines a subclass of the type \(ll\) quadratic systems with limit cycles in the ``Chinese classification''. The number and position of the limit cycles of \(C_{ll}\) is not known, although there is reason to believe that the system has at most two limit cycles. Some delicate properties of the bifurcation structure of the system in \((\ell,\delta,m)\) parameter space are considered in the paper under review. In particular, the author determines some of the surfaces given by \(\delta:= \delta(l,m)\) on which \(C_{ll}\) has either a multiple limit cycle or a separatrix cycle. For example, for the asymptotics given by \(m\) fixed and \(\ell\to\infty\), it is shown that, for \(\epsilon:= 1/\ell\), the curves in the \((\epsilon,\delta)\) space corresponding to systems with a multiple limit cycle or with a separatrix cycle have the same asymptote near \(\epsilon= 0\), in fact, in both cases, \(\delta(\ell,m)= (2- m)/4\ell+ o(1/\ell)\). This and several other results are obtained from a nontrivial analysis using the properties of quadratic systems and singular perturbation theory. As a corollary, it follows that at least two limit cycles can be created in the class of quadratic systems from an invariant parabola, from a cyclic formed by the equator of the Poincaré sphere and an invariant line, or from a separatrix cycle formed by two critical points at infinity, one of them an elementary saddle, the other a degenerate critical point.
scientific article; zbMATH DE number 665705