On entry–exit formulas for degenerate turning point problems in planar slow–fast systems

In this paper, we study degenerate entry–exit problems associated with planar slow–fast systems having an invariant line { ( x , y ) : y = 0 } with a turning point at x = 0 . The degeneracy stems from the fact that the slow flow has a saddle-node of even order 2 n , n ∈ N , at the turning point, i.e. x ′ = − x 2 n ( 1 + o ( 1 ) ) for ϵ = 0 . We are motivated by the appearance of such turning point problems (for n = 1 ) in the graphics ( I 2 1 ) and ( I 4 1 ) , through a nilpotent saddle-node singularity at infinity, in the Dumortier-Roussarie-Rousseau program (for solving the finiteness part of Hilbert’s 16th problem for quadratic polynomial systems). Our results show, under additional hypothesis, that in the case n = 1 there is a well-defined entry–exit relation for ϵ → 0 . The associated Dulac map is smooth w.r.t. ( ϵ , ϵ log ⁡ ϵ − 1 ) . On the other hand for the cases