3,779 research outputs found

    On isolation of singular zeros of multivariate analytic systems

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    We give a separation bound for an isolated multiple root xx of a square multivariate analytic system ff satisfying that an operator deduced by adding Df(x)Df(x) and a projection of D2f(x)D^2f(x) in a direction of the kernel of Df(x)Df(x) is invertible. We prove that the deflation process applied on ff and this kind of roots terminates after only one iteration. When xx is only given approximately, we give a numerical criterion for isolating a cluster of zeros of ff near xx. We also propose a lower bound of the number of roots in the cluster.Comment: 17 page

    Structure of characteristic Lyapunov vectors in spatiotemporal chaos

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    We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in systems with spatiotemporal chaos. We focus on characteristic LVs and compare the results with backward LVs obtained via successive Gram-Schmidt orthonormalizations. Systems of a very different nature such as coupled-map lattices and the (continuous-time) Lorenz `96 model exhibit the same features in quantitative and qualitative terms. Additionally we propose a minimal stochastic model that reproduces the results for chaotic systems. Our work supports the claims about universality of our earlier results [I. G. Szendro et al., Phys. Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.Comment: 9 page

    An inverse Sturm-Liouville problem with a fractional derivative

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    In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order α(1,2)\alpha\in(1,2) of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of Computational Physic
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