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We investigate bifurcation scenario of the von Karman equations with partially clamped boundary conditions defined in rectangular domains. First, we study how the (preconditioned) Block GMRES method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations. Next, we discuss linear stabilities of the von Karman equations with partially clamped boundary conditions. In particular, we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von Karman equations via computer algebra. The Block GMRES method is used to solve linear systems and to detect singularity along solution paths of the discrete problem. Sample numerical results are reported. (C) 2001 Elsevier Science B.V. All rights reserved

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Recommended by R S MacKayAbstrad We derive and analyse anormal form governing dynamics of Hopf bifurwtions of paiial differential evolution equations on a square domain. We assume that the differential optmtor for the linearized problem decomposes into two one-dimensional self-adjoint operators and a local 'reaction' operator; this gives a basis of i.e. of the form XI, xz) = fi (XI) fz(xz).The nor& form reduces to that invesfigated by Swift [23] for bifurcation of modes with odd parity but is new for modes with even parity where the centre eigenspace carries a reducible action of 0 4 x St. We consider the Brusselator equations as an example and discover that a separable linearization introduces a degeneracy which causes the three: new third order terms in the normal form to be related in an unexpected but simple way,.AMS classification scheme numbers: 35832, 58F14, 58F35.58628. 65G15