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Show constant. The second, ~, is the ratio of the heat of combustion to the specific heat (at constant pressure). The two groups are: :J = (vc/uo ) (ko) (Mt I22.4)n-l(Mt)1-a (Mox)-b (gs)b Eq.16 Eq.17 SOLUTION METHOD OF BIFURCATION ANALYSIS The solution method of a bifurcation analysis involves tracking the steady state solutions of a dynamical system as the bifurcation parameter, which for our case is 4>, is varied. The first order three dimensional system, Eq. 10, Eq. 11, and Eq. 12 were obtained by setting the time derivatives to zeros and solving the residual equations, using Newton's method. This then yielded a set of steady-state points as a function of 4>, which then described curves in the 9,11f, 110x - 4> space. The stability of the steady state points were determined by using the standard approach of linearizing the equations, to determine the Jacobian, and then determining the eigenvalues of the system. The sign of the real part of the eigenvalues determine whether or not the solutions will exponentially grow (positive eigenvalues) or decay (negative eigenvalues). A change in stability occurs when negative real eigenvalues passes through zeros and becomes positive as 4> changes. Additionally a complex conjugate pair of eigenvalues can cross the imaginary axis, and oscillatory behavior will occur when the real part of a pair of eigenvalues is zero, while the imaginary part are non-zeros complex conjugates. Hopf bifucation theory, then implies the existence of a limit cycle, or a periodic solution, with a determinable frequency. RESULTS The bifucation analysis resulted in a set of points, (9,11(, 11ox;4» which represents the steady-state solutions. The points, when graphed describe a closed loop, in 4-dimensional space. Figure 1 illustrates the key results of the bifurcation analysis, and is limited to the 7 |