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The stability of the steady states of a flow-type chemical reactor has been analyzed previously in [1]. The adiabatic conditions of the process made it possible to reduce the system of two partial differential equations for temperature and concentration to one equation for temperature. In the general case, with allowance for heat losses, which always occur in actual objects, the problem remains essentially two-dimensional, and this leads to a qualitatively new result: bifurcation of steady solutions gives rise to solutions that are periodic in time. The qualitative difference between the solutions of the adiabatic and polytropic problems remains in the case of an ideal-mixing reactor [2]. Indeed, bifurcation of a steady state to a periodic-in-time flow is typical of many-dimensional problems formulated as systems of both ordinary differential equations and partial differential equations. In this case, a one-dimensional (in temperature) nonlinear problem formulated in infinite-dimensionai space can have, along with steady solutions, periodic solutions, which result from the secondary bifurcation due to the nonlinearity of the law of heat release. This is the case when the solution of an infinite-dimensional problem is attracted to finite-dimensional space of dimension i> 2. If the solution is attracted to two-dimensional space, periodic solutions develop if the double real root of the characteristic cubic equation obtained in [1] splits into a pair of complex conjugate roots. The stability conditions for periodic-in-time solutions that result from secondary bifurcation are presented in [3].According to the procedure adopted in [1], the stability of a polytropic chemical reactor is studied by the projection method [4], although the problem considered can be reduced to the central variety. The latter method was applied to analysis of the cycle-birth bifurcation in a "brusselator" [5]. The absence of quadratic terms in unknown functions in the nonlinear mathematical model of a "brusselator" simplifies an analysis considerably. However, in spite of this, application of the theorem on the central variety brings about, in our opinion, more cumbersome calculations in comparison with the projection method.In the statement of the problem, all parameters are initially given in dimensional quantities, and the functions are specified. This is done to narrow the area of stability investigation, which follows from the physical restrictions on the parameters and their functions.1. The operation of a polytropic chemical reactor is described by the following system of differential equations [6]: OT(zh q)/Otl = ~eO2T(zl, tl)/Oz 2 -wlOT(xl, tl)/Oxl + Qkoc;l~(T, c) -~1 (T);(1.1) Oc(x:, tl)/Ot: = O02c(x:, t:)/Ox 2 -wlOT(zl, t:)/0Xl -ko~(T, c).( 1.2) Here xl is the coordinate; tl is time; T is the temperature; ae is the thermal diffusivity; c is the concentration; Q is the reaction-heat release per unit mass; E is the activation energy; R is the universal gas constant; k0 is the preexponential factor; cp is the specific heat; wl is the...

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An approximate solution ω = A [ω, μ] of the nonlinear integral Nekrasov equation is obtained by successive replacement of the kernel of the integral operator by a close one. The solution is sought not directly at the bifurcation point μ 1 = 3 of the linearized equation ω = μL[ω] but at the point μ = 1 at which operator A [ω, μ], remaining nonlinear in ω, is linear in μ.