High-order Bisection method for computing invariant manifolds of two-dimensional maps

Roy H. Goodman, Jacek K. Wróbel

Research output: Contribution to journalReview articlepeer-review

14 Scopus citations


We describe an efficient and accurate numerical method for computing smooth approximations to invariant manifolds of planar maps, based on geometric modeling ideas from Computer Aided Geometric Design (CAGD). The unstable manifold of a hyperbolic fixed point is modeled by a piecewise Bézier interpolant (a CatmullRom spline) and properties of such curves are used to define a rule for adaptively adding points to ensure that the approximation resolves the manifold to within a specified tolerance. Numerical tests on a variety of example mappings demonstrate that the new method produces a manifold of a given accuracy with far fewer calls to the map, compared with previous methods. A brief introduction to the relevant ideas from CAGD is provided.

Original languageEnglish (US)
Pages (from-to)2017-2042
Number of pages26
JournalInternational Journal of Bifurcation and Chaos
Issue number7
StatePublished - Jul 2011

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics


  • Invariant manifold
  • computer-aided geometric design
  • numerical algorithm


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