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Comment on "Identification of low order manifolds: Validating the algorithm of Maas and Pope" [Chaos 9, 108–123 (1999)]

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons86167

Flockerzi,  D.
Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons86319

Heineken,  Wolfram
Process Systems Engineering, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society;

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Citation

Flockerzi, D., & Heineken, W. (2006). Comment on "Identification of low order manifolds: Validating the algorithm of Maas and Pope" [Chaos 9, 108–123 (1999)]. Chaos, 16(4), 048101. doi:10.1063/1.2357332.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-9B36-7
Abstract
In [7] it is claimed by Rhodes, Morari and Wiggins that the projection algorithm of Maas and Pope [6] identifies the slow invariant manifold of a system of ordinary differential equations with time scale separation. A transformation to Fenichel normal form serves as a tool to prove this statement. Furthermore in [7] it is conjectured that away from a slow manifold the criterion of Maas and Pope will never be fulfilled. We present two examples that refute the assertions of Rhodes, Morari and Wiggins. In the first example the algorithm of Maas and Pope leads to a manifold that is not invariant but close to a slow invariant manifold. The claim of Rhodes, Morari and Wiggins that the Maas and Pope projection algorithm is invariant under a coordinate transformation to Fenichel normal form is shown to be not correct in this case. In the second example the projection algorithm of Maas and Pope leads to a manifold that lies in a region where no slow manifold exists at all. This rejects the conjecture in [7] mentioned above. © 2006 American Institute of Physics [accessed 2013 August 16th]