The authors consider a flow with a circular chain-recurrent set. The aim of the paper is to describe the connection between the topology of the attraction domain of an attractor and the dynamics of the flow on the attractor via the Reidemeister torsion. They show that the Reidemeister torsion of a level surface of a Lyapunov function and of the attraction domain of an attractor may be expressed in terms of the twisted Lefschetz zeta function built via closed orbits in the attractor. In the sequel they show that the Nielsen zeta function has positive radius of convergence. They give a lower estimate of the radius by means of the Reidemeister trace formula for generalized Lefschetz numbers.