Algebraic equations bifurcations

Application of computer analytical methods. Extensive use of computer analytic methods are thought to intensify theoretical analysis drastically. They will be applied, in particular, to study kinetic models of complex reactions that can be represented by systems of non-linear algebraic equations, for the detailed bifurcation analysis, etc. 

Elimination of x and X from these three equations gives a single algebraic equation in , defining a hypersurface. When values cross the H variety two bifurcation points appear or disappear and the nature of the bifurcation diagram changes as shown in Figure l.a. 

When mixture states are computed from a volumetric equation of state, then instabilities can be related to bifurcations in an algebraic equation, just as we found for pure fluids in 8.2. Inversely, if no bifurcations occur, then the mixture remains a stable single phase over all compositions, and the fugacity is a smooth monotonically increasing curve, as shown for 100 bar in Figure 8.13. Analogous behavior is observed for g xi) the stability requirement (8.4.6) on the second derivative of g " defines a simple convex curve for g (xi), like that shown on the left in Figure 8.14. 

Let us assume that, for physical reasons, we are seeking only real solutions of (19.2.1). When < 0 there is only one real solution, but when X. > 0 there are three solutions, as shown in Fig. 19.1. The new solutions for X > 0 branch or bifurcate from the solution a = 0. The value of X at which new solutions bifurcate is called the bifurcation point. In Fig. 19.1 X = 0 is the bifurcation point. Similar bifurcation of new solutions from a given solution occurs generally in nonlinear equations, be they a simple algebraic equation as above, a set of coupled ordinary differential equations or more complex partial differential equations. 

These equations can be made dimensionless by first choosing an appropriate scaling of the time variable, say, t = xlk. Whereas dimensionless equations are not necessary for carrying out a stability analysis, they often simplify the associated algebra, and sometimes useful relationships between parameters that would not otherwise be readily apparent are revealed. It is also important to note that the particular choice of dimensionless variables does not affect any conclusions regarding number of steady states, stability, or bifurcations in other words, the dimensionless equations have the same dynamical properties as the original equations. Introducing the definition t = into the above equations we find ... 

AUT097 (http //indy.cs.concordia.ca/auto) A code for tracking by continuation the solution of systems of nonlinear algebraic and/or first-order ordinary differential equations as a function of a bifurcation parameter (available only for UNIX-based computers)... 

The systems discussed above belong to the class of non-linear dynamical systems. In such cases, the non-linear equation represents evolution of a solution with time or some variable like time. Such non-linear equations may be (i) algebraic, (ii) functional, (iii) ordinary partial differential equations and (iv) integral equations or a combination of these. Non-equilibrium systems can be defined by the type of equations as defined above involving the bifurcation parameter. The solution may change depending on the particular values of parameter. The solutions change at bifurcation points. Such situations do occur in the form of bistability and oscillations. 

Because the defining conditions can be solved for the state variables and two parameters e.g. f and q), the above mentioned varieties are said to be of codimension-2. The dynamic model of the reactive flash contains several algebraic, but only one differential equation, when the holdup and pressure are fixed and the phase equilibrium is instantaneous. Such one-dimensional systems cannot exhibit Hopf bifurcations leading to oscillatory behavior. Therefore, dynamic classification is not necessary. 

Equation (7.15) predicts that the derivative of the solution with respect to the parameter will approach infinity as b + 2x approaches zero. Intuitively, as illustrated in Figure 7.9, it makes sense that an infinitely rapid change in the derivative would correspond to a qualitative change in the character of the solution. It is indeed the case that an infinite derivative indicates a bifurcation. We shall see shortly how to follow x asb changes, but a bit of algebra will tell us immediately what we want to know. We can rearrange eq. (7.13) in the form... 

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