By Eugene L. Allgower, Klaus Böhmer, Mei Zhen (auth.), Prof. Dr. Eugene L. Allgower, Prof. Dr. Klaus Böhmer, Prof. Dr. Martin Golubitsky (eds.)

Symmetry is a estate which happens all through nature and it truly is for this reason usual that symmetry could be thought of whilst trying to version nature. in lots of situations, those versions also are nonlinear and it's the learn of nonlinear symmetric versions that has been the foundation of a lot contemporary paintings. even supposing systematic reports of nonlinear difficulties should be traced again not less than to the pioneering contributions of Poincare, this continues to be a space with difficult difficulties for mathematicians and scientists. Phenomena whose versions show either symmetry and nonlinearity bring about difficulties that are demanding and wealthy in complexity, attractiveness and application. in recent times, the instruments supplied through staff conception and illustration thought have confirmed to be powerful in treating nonlinear difficulties related to symmetry. through those capacity, hugely advanced occasions could be decomposed right into a variety of less complicated ones that are already understood or are no less than more straightforward to address. within the realm of numerical approximations, the systematic exploitation of symmetry through workforce repre­ sentation idea is much more contemporary. within the wish of stimulating interplay and acquaintance with effects and difficulties within the numerous fields of purposes, bifurcation concept and numerical research, we prepared the convention and workshop Bifurcation and Symmetry: move affects among arithmetic and functions in the course of June 2-7,8-14, 1991 on the Philipps­ college of Marburg, Germany.

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1. 1: Let (Ao,d o ) E S(LcJ n S(Lc,), where Cl # C2 represent It and 12 fold eigenvalues of the Laplacian L\ on n with Dirichlet boundary conditions, respectively. 22) as i, i = 1, ... , Ii, j = 1,2. 4) 54 K. Bohmer and Z. Mei 4. ,d)) jumps at the intersection points of S(L C1 ) with another bifurcation curve S(Lc,). New bifurcating solution branches, or the so-called mode interactions are expected at these points (cf. [8], [11]). For reasons of simplifications we confine the discussions to the special =f 0 and to the points (>'0, do) E S(L C1 ) n S(Lc,) with case ifu, .

So that < i, Axo > is positive to ensure that the additional eigenvalues lie on the real axis. This is achieved by choosing e = Axo where Xo is the last solution found. Detection of this type of Hopf bifurcation can be achieved by using a package such as AUTO (Doedel (1981)) on the system G(y, A) = O. We note that on a travelling wave solution of G(y, A) = 0, x is independent of time and so x = x. , x >= 0 which is a simple algebraic equation that is easily implemented. A more efficient method of detection is to use the block diagonalisation of Gy(y, A) which arises due to the symmetry of the problem.

Thus, under the assumption that all eigenspaces are spanned by a finite set of Fourier modes, the dimension of the kernel is equal to the number of solutions of: m _Z22 + _ 2 2 b a = Ikl 2 = (P-l(0))2 (1) for (I, m) E Z2. /d) with integers p, q and some square-free integer d > 0, means we can rewrite equation 1 as q2Z2 + dp 2m 2 = n, and thus the number of Writing alb solutions is at least R(nl(pq)2) for any n such that pqly'n, where R(n) is the number of solutions of the equation with dEN square-free and (z,y) E Z2.

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