Abstract
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 language | English (US) |
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Pages (from-to) | 2017-2042 |
Number of pages | 26 |
Journal | International Journal of Bifurcation and Chaos |
Volume | 21 |
Issue number | 7 |
DOIs | |
State | Published - Jul 2011 |
All Science Journal Classification (ASJC) codes
- Modeling and Simulation
- Engineering (miscellaneous)
- General
- Applied Mathematics
Keywords
- Invariant manifold
- computer-aided geometric design
- numerical algorithm