Bifurcation problems

Chan, T. (1984). in Numerical Methods for Bifurcation Problems, T. Kup-per, H. Mittelmann and H. Weber, Eds., Birkhauser Verlag, Basel. 

Guckenheimer, J. (1986). Multiple bifurcation problem for chemical reactors. Physica, D20, 1-20. 

Specifically, in Chapter 3 we create a surface for a transcendental function /(a, y) as an elevation matrix whose zero contour, expressed numerically as a two row matrix table of values, solves the nonlinear CSTR bifurcation problem. In Chapter 6 we investigate multi-tray processes via matrix realizations in Chapter 5 we benefit from the least squares matrix solution to find search directions for the collocation method that helps us solve BVPs and so on. Matrices and vectors are everywhere when we compute numerically. That is, after the laws of physics and chemistry and differential equations have helped us find valid models for the physico-chemical processes. 

Doedel, E., Keller, H., Kemevez,). Numerical analysis and control of bifurcation problems (I) Bifurcation in finite dimensions. Int.J. Bif. Chaos 1991,1 493-520. 

The shortcut used here exploits the fact that /(x, y3) = -x + 3 tanh x depends more simply on y3 than on x. This is frequently the case in bifurcation problems— the dependence on the control parameter is usually simpler than the dependence on X. ... 

Continuation and bifurcation problems Computational fluid dynamics... 

We shall examine the class of bifurcation problems for which the system (5.58) has the stationary state (x0 c0), that is f(x0 c0) = 0, for which the condition Re ) =. .. = Re(Ap) = 0 is satisfied. Such a stationary state is sensitive. As we have established previously, a catastrophe involving a change in the phase portrait near a stationary point takes place on the system crossing a sensitive state in which the stability matrix has the eigenvalue (eigenvalues) equal to zero or a pair (pairs) of purely imaginary eigenvalues. 

J. Guckenheimer, Multiple bifurcation problems of codimension two , SIAM J. Math. Anal., 15, 1 (1984). 

Doedcl, E, 1986, Auto software for continuation and bifurcation problems in ordinary differential equations, California Inst, of Tech., Pasadena, California, USA. 

Doedel, E. AUTO Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Dept, of Mathematics, California Institute of Technology, Pasadena, CA, 1986. 

A multilayered feed forward neural network with input, hidden and output layer is chosen. The choice follows recommendation of Hurtado Alvares (2001), which argue that radial basic functions (RBF) networks are not suitable for bifurcation problems. 

Stochastic bifurcation problems in chemical systems have been analysed by Lemarchand (1980). He proposed a systematic expansion of the free-energy-like quantity ( stochastic potential ) in powers of... 

Doedel, E. Keller, H. B. Kernevez., J. 1991. Numerical Analysis and Control of Bifurcation Problems (I) Bifurcation in Finite Dimensions, Int. J. Bifurc. Chaos 1, 493-520. 

In Section 2.6.1, a model bifurcation problem was considered for the departure from axially symmetric deformation of a circular film-substrate system due to increasing mismatch strain in the film. Consider the same basic problem except that the shape of the substrate is a square of extent L on a side, rather than a circle of radius R. The undeformed substrate midplane coincides with the plane, and the coordinate axes are perpendicular to the sides of the square. 

W. F. Langford. Proceeding of the conference on Numerical Methods for Bifurcation Problems.Univ.Dortmund(1983)... 

Sey84] Seydel R. (1984) A continuation algorithm with step control. In Kiipper T. (ed) Numerical Methods in Bifurcation problems, volume 70 of ISNM, pages 480-494. Birkhauser. 

In this presentation BF coordinates are chosen because they lead to simpler final expressions for the cross sections and to a simpler solution to the bifurcation problem. However, SF coordinates can also be used [121]. In terms of these BF coordinates, we wish to find the solutions of the Schrbdinger equation ... 

The basic state then becomes unstable with respect to steady non-uniform modes in x and y and we observe a transition to stationary cellular flames. Since the instability is preferential with respect to zero wave number perturbations, the bifurcation problem can be simplified by seeking new evolution equations valid for small wave number perturbations. This analysis leads to a nonlinear equation for the flame shape (fiCx.y.t) = given by... 

We now analyze equation (3.1) in detail. Our purpose is to show that the interaction of steady modes may lead to a bifurcation to time-periodic solutions. The problem is motivated by the observation that polyhedral flames on a Bunsen burner may sometimes rotate about its vertical axis. Similar bifurcation problems were recently discussed in the context of convective instabilities [9]. Here we analyse the simplest case of a two-dimensional (z ,y) flame subject to zero-flux boundary conditions in the y-direction ... 

In the general case where there are both stable and unstable characteristic exponents, or stable and unstable multipliers in the spectrum, the local bifurcation problem does not cause any special difficulties, thanks to the reduction onto the center manifold. Consequently, the pictures from Chaps. 9-H will need only some slight modifications where unstable directions replace stable ones, or be added to existing directions in the space. However, the reader must... 

In principle, to solve a bifurcation problem we need to consider all systems close to XsQ, This means that we must consider the Banach space of all small perturbations." On the other hand, when it is possible to reduce the analysis to some appropriate finite-parameter family of systems, the study is simplified significantly. 

The tool kit used for studying bifurcational problems consists of three pieces the theorem on center manifold, the reduction theorem, and the method of normal forms. 

The computing algorithms of most of these bifurcations have been well developed and can therefore be implemented in software we mention here the packages designed to settle these bifurcation problems LOCBIF [76], AUTO [46] and CONTENT [83]. The exception is the blue sky catastrophe, Despite the fact that it is a codimension-one boundary, this bifurcation has not yet been found in applications of nonlinear dynamics although an explicit mathematical model does exist [53]. 

Krauskopf, B. and Osinga H. M. [1999] Investigating torus bifurcations in the forced Van der Pol oscillator , in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems eds. E. J. Doedel, L. S. Tuckerman, IMA Volumes in Mathematics and its Applications 119, Springer-Verlag, to appear. 

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