Finite cyclicity of slow-fast Darboux systems with a two-saddle loop (Q2817000)

Let \(P_{0}=y-x^{2}, P_{1}=1-y\) and \(P(x,y,\delta), Q(x,y,\delta)\) be real polynomials depending analytically on \(\delta\), \(P(x,y,0)=Q(x,y,0)=0\). Consider the system NEWLINE\[NEWLINE \dot{x}= P_{0}-\epsilon P_{1}-Q(x,y,\delta),\,\dot{y}=-2 \epsilon x P_{1} +Q(x,y,\delta)NEWLINE\]NEWLINE underlying the foliation NEWLINE\[NEWLINE F_{\epsilon,\delta}: \epsilon P_{1} dP_{0} +P_{0} d P_{1}+ Pdx+Qdy=0.NEWLINE\]NEWLINE For every small positive \(\epsilon\), \(Z(\epsilon)\) denotes the maximal number of limit cycles of \(F_{\epsilon,\delta}\), which bifurcate from the region bounded by \(y=x^{2}, y=1\) for small \(\delta\). The main result of this paper is that \(Z(\epsilon)\) is finite and uniformly bounded in \(\epsilon >0\).