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Solution (solid lines) and intrinsic low-dimensional manifold (dashed lines) of the second example.
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It is claimed by Rhodes, Morari, and Wiggins [Chaos9, 108–123 (1999)] that the projection algorithm of Maas and Pope [Combust. Flame88, 239–264 (1992)] 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, Rhodes, Morari, and Wiggins [Chaos9, 108–123 (1999)] 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 of Rhodes, Morari, and Wiggins mentioned above.
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