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Feasibility study by interval arithmetics 1052. Mathematical formulationIn all the studied variants, the differential equations have the form ofdxdh = V L (y(x) − y∗ (x)) (1)where x is the array of independent variables, h is a running variable irrelevant for us, ratioV/L is a simple function of R and F/V or F/L, y ∗ (x) is a non-explicit function describedbelow, and y(x) is an explicit function expressed in a general form asy =( RR + 1 + F )x + 1V R + 1 x D − F V x F (2)Here R is one of the main parameters of the process, and F/V is another one. x D and x F areknown process specifications. Array y ∗ is determined according to the following system ofequations (3 to 7):y ∗ i P = γ ix i p ◦ i (i = 1, 2, 3) (3)ln γ i =lg p ◦ i = A i −∑j τ jiG ji x j∑l G + ∑lix ljB iT − 273.14 + C i(i = 1, 2, 3) (4)(x j G∑ ijl G τ ij −ljx j∑n τ )njG nj x∑ nl G ljx j(i = 1, 2, 3) (5)τ ij =U ijR G T; G ij = exp (−α ij τij) (i = 1, 2, 3) (6)3∑yi ∗ = 1 (7)i=1Here P is the systems pressure, R G is a universal physical constant, A i , B i , C i , U ij , and α ijare model parameters. p ◦ i and γ i can be considered as functions of x and T ; such a T is to befound for a given x that satisfies equation (7). The problem of finding the singular points canbe formulated as solving the system of equations (2) to (8).0 = y(x) − y ∗ (x) (8)The problem of finding bifurcations can be formulated as follows. Differencial equation (1)is linearized around the singular point in a formdxdh= Ax (9)The task is to find that parameters at which at least one of the eigenvalues of A has zeroreal part. In this problem class we encounter node and saddle singularities only; thus, bifurcationsare simply signalled by zero determinant of A. Unfortunately, entries of A (theJacobian) cannot by directly determined because of the non-explicite nature of function y ∗ (x).This difficulty is solved by applying implicit function theorem. Computation of the partialderivatives of γ are also rather tedious.