Bifurcation theory

Golubitsky M and Schaeffer D G 1985 Singularities and Groups in Bifurcation Theory vol 1 (New York Springer)... 

This Is a didactic Introduction to some of the techniques of bifurcation theory discussed In this article. 

Kuhicek, M., and M. Marek. Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, Berhn (1983). 

The approach used in these studies follows idezus from bifurcation theory. We consider the structure of solution families with a single evolving parameter with all others held fixed. The lateral size of the element of the melt/crystal interface appears 2LS one of these parameters and, in this context, the evolution of interfacial patterns are addressed for specific sizes of this element. Our approach is to examine families of cell shapes with increasing growth rate with respect to the form of the cells and to nonlinear interactions between adjacent shape families which may affect pattern formation. 

Keller, H. B. Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems, In Applications of Bifurcation Theory, Rablnowltz,P., Ed. Academic Press New York, 1977. 

Keller, H. B. (1977). in Applications of Bifurcation Theory, P. Rabinowitz, Ed., Academic Press, New York. 

Robert A. Brown is Warren K. Lewis Professor of Chemical Engineering and Provost at the Massachusetts Institute of Technology. He received his B.S. (1973) and M.S. (1975) from the University of Texas, Austin, and his Ph.D. from the University of Minnesota in 1979. His research area is chemical engineering with specialization in fluid mechanics and transport phenomena, crystal growth from the melt, microdefect formation in semiconductors and viscoelastic fluids, bifurcation theory applied to transitions in flow problems, and finite element methods for nonlinear transport problems. He is a member of the National Academy of Engineering, the National Academy of Sciences, and the American Academy of Arts and Sciences. 

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, Berlin (1995). 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

Chain reactions and explosions are also autocatalytic processes. See also Bifurcation Theory Prion Plaque Formation... 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

Slow transitions produced by enzyme isomerizations. This behavior can lead to a type of cooperativity that is generally associated with ligand-induced conformational changes . A number of enzymes are also known to undergo slow oligomerization reactions, and these enzymes may display unusual kinetic properties. If this is observed, it is advisable to determine the time course of enzyme activation or inactivation following enzyme dilution. See Cooperativity Bifurcation Theory Lag Time... 

BIFURCATION THEORY PRION PLAQUE FORMATION Autocatalytic processes during evolution, HYPERCYCLE AUTOINHIBITION ACTIVATION... 

BIFUNCTIONAL CATALYSIS BIFUNCTIONAL ENZYME BIFURCATION THEORY BILIVERDIN REDUCTASE BIMOLECULAR... 

BIFURCATION THEORY JENCKS CLOCK JENKINS MECHANISM "JMP,"... 

In the preceding sections we have analyzed the new solutions that appear at a point of instability and have shown that they can be calculated by the methods of bifurcation theory as long as their amplitude is small. In this section we consider a system of the form (2) whose steady-state solutions can be evaluated straightforwardly without implying any restriction on the parameters value. This allows a complete analysis of these branches of solutions.32... 

These results are thus in agreement with those of bifurcation theory. In the case of odd wave numbers they demonstrate that in general the bifurcation diagrams have to exhibit a subcritical branch. However, there always exists even for odd wave numbers a value of the parameters such that the bifurcation is soft and this value marks the transition from an upper to a lower subcritical branch (see Fig. 21). This feature was less... 

The mathematical theory of dissipative structures is mainly based on approximate methods such as bifurcation theory of singular perturbation theory. Situations like those described in Section VI and that permit an exact solution are rather exceptional. 

Iooss, G. and Joseph, D. D. (1980). Elementary stability and bifurcation theory. Springer, New York. 

A. Andronov, E. Leontovich, I. Gordon and A. Mayer, Bifurcation Theory of Planar Dynamical Systems (Nauka, Moscow, 1967). 

Chang, H. C., 1983, The domain model in heterogeneous catalysis. Chem. Engng ScL 38,535-546. Cohen, D. S. and Neu, J. C., 1979, Interacting oscillatory chemical reactors. In Bifurcation Theory and Applications in Scientific Disciplines. N.Y. Acad. Sci., New York. 

Kubicek, M. Marek, M. 1983 In Computational methods in bifurcation theory and dissipative structures, pp. 39-47. New York Springer-Verlag. 

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