NP-hard combinatorial optimization problems are in general hard problems that their computational complexity grows faster than polynomial scaling with the size of the problem. Thus, over the years there has been a great interest in developing unconventional methods and algorithms for solving such problems. Here, inspired by the nonlinear optical process of $q$-photon down-conversion, we introduce a single-variable nonlinear dynamical model that allows for efficiently approximating the ground state of the classical $q$-state planar Potts Hamiltonian. This reduces the exhaustive search in the large discrete solution space of a large class of combinatorial problems that are represented by the Potts Hamiltonian to solving a system of coupled dynamical equations. We introduce two different annealing mechanisms, adiabatic deformation and chaotic annealing, and show that numerical simulations of the proposed dynamical model can efficiently approximate the solution of hard combinatorial problems. The proposed algorithm is applied to graph-$q$-partitioning as an example.