We deal with the large-time behaviour of scalar hyperbolic conservation laws with source terms
which are often called hyperbolic balance laws. Fan and Hale have proved existence of a global attractor for this equation with x ∈ S1. consists of spatially homogeneous equilibria, a large number of rotating waves and of heteroclinic orbits between these objects. In this paper, we solve the connection problem and show which equilibria and rotating waves are connected by a heteroclinic orbit. Apart from existence results, our approach via generalized characteristics also gives geometric information about the heteroclinic solutions, e.g. about the shock curves and their strength.