This paper studies the homotopy invariant cat(X,ξ) introduced by the first author in . Given a finite cell-complex X, we study the function ξ cat(X, ξwhere ξ varies in the cohomology space Hl (X: R). Note that cat(X. ξ) turns into the classical Lusternik-Schnirelmann category cat(Z) in the case ξ = 0. Interest in cat(X ξ) is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1-forms, see [6; 7]. In this paper we significantly improve earlier cohomological lower bounds for cat(X. ξ) suggested in [6; 7]. The advantages of the current results (see Theorems 5, 6 and 7 below) are twofold: firstly, we allow cohomology classes ξ of arbitrary rank (while in  the case of rank one classes was studied), and secondly, the theorems of the present paper are based on a different principle and give slightly better estimates even in the case of rank one classes. We introduce in this paper a new controlled version of eat(X ξ) and find upper bounds for it (Theorems 11 and 16). We apply these upper and lower bounds in a number of specific examples where we explicitly compute cat(X, ξ) as a function of the cohomology class ξ ε H1(X; ℝ).