We study a class of dynamical systems on a torus that includes dynamical systems modeling the dynamics of the Josephson transition. For systems in this class, we introduce certain characteristics including a sequence of functions depending on the system parameters. We prove that if this sequence converges at a given point in the parameter space, then its limit is equal to the classical rotation number, and we then call this point a quantization point for the rotation number. We prove that the rotation number of such a system takes only integer values at a quantization point. Quantization areas are thus defined in the parameter space, and the problem of effectively describing them becomes an important part of characterizing the systems under study. We present graphs of the rotation number at quantization points and under conditions when it is not quantized (an example of a half-integer rotation number) and diagrams for quantization areas. Keywords: dynamical system on a torus, rotation number, quantization, Josephson effect According to the classical Poincaré-Denjoy [1]-[3] theory, the solution behavior of a system of differential equations of the formẋwhere µ is a real constant and G(x+2π, y) = G(x, y+2π) = G(x, y) and F (x+2π, y) = F (x, y+2π) = F (x, y) are smooth functions, is characterized by a homeomorphic map J : C → C of the circle C onto itself and by the rotation number ν. In the case µ → 0, system (1) relates to the class of the so-called slow-fast systems of differential equations on a torus [4]- [6]. It was shown in [5] that for G(x, y) = 1 and F (x, y) = a−cos x−cos y, system (1) in the case µ → 0 has some specific properties noncharacteristic of generic systems on the plane. Our objective here is to show that system (1) of the forṁwhereand g(y + 2π) = g(y), 2π 0 g(y) dy = 0, are smooth functions, a 0 is a real constant, and M is a point in the parameter space R of the function f (x), has some properties noncharacteristic of generic systems on a torus. As we show below, the characteristics J and ν are insufficient for describing systems (2). We introduce some new characteristics, namely, the quantization number m(µ, a 0 , M), the function λ(µ, a 0 , M), and the function i s (µ, a 0 , M).