2011
DOI: 10.1142/s0218127411029604
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High-Order Bisection Method for Computing Invariant Manifolds of Two-Dimensional Maps

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 Catmull-Rom 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… Show more

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Cited by 19 publications
(13 citation statements)
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“…We refer the reader to the works of [25,23,22,21,20,19,42,43] as well as to the review article [18]. The paper [24] treats the globalization of non-orientable stable/unstable manifolds of periodic orbits in differential equations, a subject which is also treated using the techniques of the present work.…”
Section: Related Workmentioning
confidence: 99%
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“…We refer the reader to the works of [25,23,22,21,20,19,42,43] as well as to the review article [18]. The paper [24] treats the globalization of non-orientable stable/unstable manifolds of periodic orbits in differential equations, a subject which is also treated using the techniques of the present work.…”
Section: Related Workmentioning
confidence: 99%
“…Numerical methods for computing invariant manifolds based on the Parameterization method can be found in a number of works including [56,57,72,42,43,82], along with the present work. A useful feature of methods based on Parameterization is that they admit natural a-posteriori error indicators.…”
Section: Related Workmentioning
confidence: 99%
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“…In [38,37] the authors treat one and two dimensional invariant manifolds of maps and develop an adaptive globalization scheme which exploits Bézier curves and triangles. A feature of this work is that the authors pre-condition their globalization scheme with a high order approximation of the local invariant manifold based on the Parameterization Method.…”
Section: Computing Invariant Manifolds: a Brief Overviewmentioning
confidence: 99%
“…For a given nonlinear dynamical system, the only general way of studying such stable and unstable manifolds is by computing them numerically. Consequently, a number of different algorithms have been developed for computing the stable and unstable manifolds for autonomous maps [1,3,6,10,16,18,25,26].…”
mentioning
confidence: 99%