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Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies

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Abstract

We illustrate and discuss the computer-assisted study (approximation and visualization) of two-dimensional (un)stable manifolds of steady states and saddle-type limit cycles for flows in R n. Our investigation highlights a number of computational issues arising in this task, along with our solutions and “quick-fixes” for some of these problems. Two examples illustrative of both successes and shortcomings of our current approach are presented. Representative “snapshots” demonstrate the dependence of two-dimensional invariant manifolds on a bifurcation parameter as well as their interactions. Such approximation and visualization studies are a necessary component of the computer-assisted study and understanding of global bifurcations.

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Authors and Affiliations

  1. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Mark E. Johnson

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Michael S. Jolly

  3. Program in Applied and Computational Mathematics and Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544, USA

    Ioannis G. Kevrekidis

Authors
  1. Mark E. Johnson
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  2. Michael S. Jolly
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  3. Ioannis G. Kevrekidis
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Johnson, M.E., Jolly, M.S. & Kevrekidis, I.G. Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies. Numerical Algorithms 14, 125–140 (1997). https://doi.org/10.1023/A:1019104828180

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