Influence of a small perturbation on Poincaré-Andronov operators with not well defined topological degree (Q1013005)

Consider the differential equation \(\dot x=f(x)+ \varepsilon g(t,x,\varepsilon)\) for small \(\varepsilon>0\) on a bounded open set \(U\) with smooth boundary. A solution of it with initial condition \(x(0)=v\) is denoted by \(x_\varepsilon(.,v)\). For \(T>0\) the map \(v\mapsto{\mathcal P}_\varepsilon(v)= x_\varepsilon(T,v)\) is called the Poincaré-Andronov operator. Assume that \(x_0\) is a \(T\)-periodic cycle for \(\varepsilon=0\). The topological degree \(d(I-{\mathcal P}_\varepsilon)\) is calculated. It is shown that there are infinitely many integers such that for small \(\varepsilon\) this degree can be any of these integers although the set of fixed points of \({\mathcal P}_0\) has Lebesgue measure 0.