Bifurcation analysis

We have stated that we do not in general know the number or even the existence of solutions to a nonlinear algebraic system. This is true however, it is possible to identify points at which the existence properties of the system change through locating bifurcation points i.e., choices of parameters at which the Jacobian, evaluated at the solution, is singular. 

Consider a simple case of two nonlinear equations whose solution(s) depend upon some parameter vector 0. 

We can search for a bifurcation point along some path 0(A.) in parameter space by solving the augmented set of iV + 1 equations for Xs( c) and the value ofA-c, (A-c) = c, at which the bifurcation occurs, 

Computing the locations of bifurcation points allows us to carve up parameter space into different regions, to find one in which one or more real solutions do indeed exist. 

Example. Bifurcation points of a simple quadratic equation 

The form of the model that will be used most often in this section is given by (9). Although there are four parameters, the two primary parameters are a and a2, for these are proportional to the partial pressure of the reactants and would be the parameters most easily varied in an experiment. 

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The couphng constants of O nt used here in the bifurcation analysis are 6 /2r... 

The asymptotic form of Qeff (corresponding to local minima) is calculated up to 0 e ) for e 0. We thus consistently neglect in our bifurcation analysis all the terms which would lead to contributions to Qeff of order O(e ), without further stating this explicitly. 

A. Ciach. Bifurcation analysis and liquid-crystal phases in Landau-Ginzburg model of microemulsion. J Chem Phys 704 2376-2383, 1996. 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

The set of the reaction-diffusion equations (78) can be solved by different methods, including bifurcation analysis [185,189-191], cellular automata simulations [192,193], or numerical integration [194—197], Recently, two-dimensional Turing structures were also successfully studied by Mecke [198,199] within the framework of integral geometry. In his works he demonstrated that using morphological measures of patterns facilitates their classification and makes possible to describe the pattern transitions quantitatively. 

F. Teymour and W.H. Ray. The dynamic behavior of continuous solution polymerization reactors-IV. Dynamic stabihty and bifurcation analysis of an experimental reactor. Chem. Eng. Sci., 44(9) 1967-1982, 1989. 

TNC.48. G. Nicolis and I. Prigogine, Thermodynamic aspects and bifurcation analysis of spatio-temporal dissipative structures, in Proceedings, Faraday Symposium Chemical Society, no. 9, Physical Chemistry of Oscillatory Phenomena, 1975, pp. 7—20. 

TNC.68. 1. Prigogine and G. Nicolis, Self-organization in nonequilibrium systems Towards a dynamics of complexity, in Bifurcation Analysis, M. Hatzewinkel, ed., Reidel, Dordrecht, 1985, pp. 3-12. 

Sanchez, A. L., A. Linan, and F. A. Williams. 1992. A bifurcation analysis of high-temperature ignition of H2-O2 diffusion flames. 24th Symposium (International) on Combustion Proceedings. Pittsburgh, PA The Combustion Institute. 1529-37. 

H. Steinriick, A bifurcation analysis of the steady state semiconductor device equation, SIAM J. Appl. Math., 49 (1989), pp. 1102-1121. 

Bifurcation analysis in the local Teorell model accounting for negative osmosis and concentration dependence of the electro-osmotic factor as prescribed by (6.4.44). The analysis should be essentially identical to that of 6.3 with the generalized Darcy s law (6.3.11) replaced by the expression... 

Equations (3.20) and (3.21) with their stationary-state solutions (3.24) and (3.25) are simple enough to provide a good introduction to some of the mathematical techniques which can serve us so well in analysing these sorts of chemical models. In the next sections we will explain the ideas of local stability analysis ( 3.2) and then apply them to our specific model ( 3.3). After that we introduce the basic aspects of a technique known as the Hopf bifurcation analysis ( 3.4) which enables us to locate the conditions under which oscillatory states are likely to appear. We set out only those aspects that are required within this book, without any pretence at a complete... 

Hopf bifurcation analysis commonly signals the onset of oscillatory behaviour. This chapter uses a particular two-variable example to illustrate the essential features of the approach and to explore the relationship to relaxation oscillations. After a careful study of this chapter the reader should be able to ... 

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

Hopf bifurcation analysis with Arrhenius model birth and growth of oscillations... 

We have already discussed the expressions resulting from a full Hopf bifurcation analysis of the thermokinetic model with the exponential approximation (y = 0). We may do the same for the exact. Arrhenius temperature dependence (y 0). Although the algebra is somewhat more onerous, we still arrive at analytical, expressions for the stability of the emerging or vanishing limit cycle and the rate of growth of the amplitude and period at... 

The Hopf bifurcation analysis proceeds as described previously, the required condition being that the trace of the Jacobian matrix corresponding to eqns (12.45) and (12.46) should become equal to zero for some stationary-state concentration given by the lower root from (12.51). (The solution with the upper root corresponds to the middle branch of stationary states for... 

We now turn to an example where full use has been made of the bifurcation analysis based on Floquet multipliers, as described in 5.4.3. 

Aluko, M. and Chang, H.-C., 1984, PEFLOQ an algorithm for the bifurcational analysis of periodic solutions of autonomous systems. Comp. Chem. Engng 8, 355-365. 

Chan, T. N., 1983, Numerical bifurcation analysis of simple dynamical systems. M. Comp. Sci. Thesis, Concordia University, Montreal. 

Doedel, E. J., 1981, AUTO a program for the automatic bifurcation analysis of autonomous systems. Cong. Num. 30, 265-284. 

Shooting method for bifurcation analysis of boundary value problems (with X. Song and L.D. Schmidt). Chem. Eng. Commun. 84,217-229 (1989). 

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