Bifurcation equations

Order arising through nucleation occurs both in equilibrium and nonequilibrium systems. In such a process the order that appears is not always the most stable one there are often competing processes that will lead to different structures, and the structure that appears is the one that nucleates first. For instance, in the analysis of the different possible structures in diffusion-reaction systems17-20 one can show, by analyzing the bifurcation equations, that there are several possible structures and some of them require a finite amplitude to become stable if this finite amplitude is realized through fluctuation, this structure will appear. In the formation of crystals (hydrates) the situation is similar the structure that is formed depends, according to the Ostwald rule, on the kinetics of nucleation and not on the relative stability. 

The amplitude a, which remains undetermined at this stage is fixed by the solvability conditions imposed on the equations for the higher order terms of the perturbation expansion (5). We arrive in this way at the following set of bifurcation equations for the normalized amplitude (3 = ta. 

Substituting into equation (12) one then obtains a system of linear equations for xj which incorporate the effect of the imperfection already at the dominant order. By working out these relations one finally arrives at the bifurcation equations for the amplitude of the dominant part of the solution, exi. 

These are called the bifurcation equations. In fact, though (19.2.1) is an equation in its own right, it is also a bifurcation equation for systems that break a two-fold symmetry. The multiplicity of solutions to (19.2.6) corresponds to the multiplicity of solutions to the original equation (19.2.4). 

The unknown amplitudes a and 3 are determined by solving a system of ordinary differential equations, called the bifurcation equations. In [4,5], it is shown that the symmetries present in the governing equations (2.1) and the eigenfunctions of equations (2.3) are reflected in corresponding symmetry properties of the bifurcation equations. Specifically, since the problem is invariant with respect to translations in T and Y, and with respect to reflections in Y, the analysis leads to bifurcation equations of the form... 

The analysis of the bifurcation equations in this case indicates that in addition to the primary states given by (2.11) and (2.12), there exists a secondary bifurcation to quasi-periodic solutions which satisfies (2.16) with P e a2 P+ bj o). The condition for its existence and its location depends on higher order terms in the bifurcation equations. This analysis is carried out by Erneux and Matkowsky in [6]. Figure 3 exhibits a typical bifurcation diagram of the amplitude as a function of X. 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Abstract. A model of the conformational transitions of the nucleic acid molecule during the water adsorption-desorption cycle is proposed. The nucleic acid-water system is considered as an open system. The model describes the transitions between three main conformations of wet nucleic acid samples A-, B- and unordered forms. The analysis of kinetic equations shows the non-trivial bifurcation behaviour of the system which leads to the multistability. This fact allows one to explain the hysteresis phenomena observed experimentally in the nucleic acid-water system. The problem of self-organization in the nucleic acid-water system is of great importance for revealing physical mechanisms of the functioning of nucleic acids and for many specific practical fields. 

Changing the constants in the SCF equations can be done by using a dilferent basis set. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of results vs. constant values are the bifurcation diagrams that are found in many explanations of chaos theory. 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

If for certain values of a parameter A in the differential equation, the qualitative aspect of the solution (i.e., the phase portrait ) of the differential equation remains the same (in other words the changes are only quantitative) such values of A are called ordinary values. If however, for a certain value A = A0 this qualitative aspect changes, such a special value is called a critical or bifurcation value. 

Therefore, the development of an open system can be described by a set of nonlinear equations that usually have solutions in equilibrium at infinity. In some cases, the solutions change their states greatly before and after the specific values of physical parameters these phenomena are called bifurcations. Figure 1 shows a simple case of bifurcation. For example, the following nonlinear differential equation is considered,... 

Kublcek, M. Marek, I. Computational Methods in Bifurcation Theory and Dissipative Structures Springer Verlag New York, 1983. Rhelnboldt, W. C. Numerical Analysis of Parameterized Nonlinear Equations Wiley Interscience New York, 1986. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

In our computational model the strain rate and the equivalence ratio are the natural bifurcation parameters. If we denote either of these parameters by a, then the system of equations in (3.2) can be written in the form... 

In flame extinction studies the maximum temperature is used often as the ordinate in bifurcation curves. In the counterflowing premixed flames we consider here, the maximum temperature is attained at the symmetry plane y = 0. Hence, it is natural to introduce the temperature at the first grid point along with the reciprocal of the strain rate or the equivalence ratio as the dependent variables in the normalization condition. In this way the block tridiagonal structure of the Jacobian can be maintained. The flnal form of the governing equations we solve is given by (2.8)-(2.18), (4.6) and the normalization condition... 

One of the interesting consequences of changing specifications is the effect on the equation structure. With formulations C and D, the occurrence matrix is symmetric. But if external flows, w, and model parameters, / , are introduced as the unknown variables, the symmetry may be destroyed. One way of preserving the local symmetry is to augment the system of equations and to bifurcate the variables in terms of state and design variables (M2). 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve... 

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. 

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