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Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. (English) Zbl 1038.34075

The authors investigate the behaviour of delay differential equations with cubic nonlinearities in the vicinity of Hopf bifurcation points by a multiple scale ansatz. Comparing their approximation with numerical results, they find good agreement and conclude that it isn’t necessary to perform center manifold reduction to remove the infinitely many stable modes. This conclusion sounds very questionable, because it is well known, that for skew symmetric nonlinearities the simple Galerkin approximation yields correct results, whereas for quadratic terms center manifold reduction – or some equivalent treatment of the stable modes – is necessary.

MSC:

34K18 Bifurcation theory of functional-differential equations
34E13 Multiple scale methods for ordinary differential equations
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