New Numerical Invariants of an Unfolding of a Polycycle "Tears of the Heart"

In this paper new numerical invariants of structurally unstable vector fields in the plane are found. One of the main tools is an improved asymptotics of sparkling saddle connections that occur when a separatrix loop of a hyperbolic saddle breaks. Another main tool is a new topological invariant of two arithmetic progressions, both perturbed and unperturbed, on the real line. For the pairs of the unperturbed arithmetic progressions we give a complete topological classification.