In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.

We consider a Hermitian matrix valued function A(x) ∈ C n×n , smoothly depending on parameters x ∈ Ω ⊂ R 3 , where Ω is an open bounded region of R 3 . We develop an algorithm to locate parameter values where the eigenvalues of A coalesce: conical intersections of eigenvalues. The crux of the method requires one to monitor the geometric phase matrix of a Schur decomposition of A, as A varies on the surface S bounding Ω. We develop (adaptive) techniques to find the minimum variation decomposition of A along loops covering S and show how this can be used to detect conical intersections. Further, we give implementation details of a parallelization of the technique, as well as details relative to the case of locating conical intersections for a few of A's dominant eigenvalues. Several examples illustrate the effectiveness of our technique.

It is well known that the nontrivial solutions of the equationblow up in finite time under suitable hypotheses on the initial data, κ and f . These solutions blow up with large oscillations. Knowledge of the blow-up profile of these solutions is of great importance, for instance, in studying the dynamics of suspension bridges. The equation is also commonly referred to as extended Fisher-Kolmogorov equation or Swift-Hohenberg equation.In this paper we provide details of the blow-up profile. The key idea is to relate this blow-up profile to the existence of periodic solutions for an auxiliary equation.

We consider matrix valued functions of two parameters in a simply connected\ud region $\Omega$. We propose a new criterion to detect when such functions\ud have coalescing singular values. For {\it generic\/} coalescings,\ud the singular values come together in a ``double cone''-like intersection.\ud We relate the existence of any such singularity to the periodic structure of the\ud orthogonal factors in the singular value decomposition of the one-parameter\ud matrix function obtained restricting to closed loops in $\Omega$.\ud Our theoretical result is very amenable to approximate numerically the\ud location of the singularities

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In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem A(x) − λB(x), where A and B are symmetric matrix valued functions in R n×n , smoothly depending on parameters x ∈ Ω ⊂ R 2 ; further, B is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where A and B are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.

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Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.