On the computation of the term $w_{21}z^2\bar{z}$ of the series defining the center manifold for a scalar delay differential equation

Details On the computation of the term $w_{21}z^2\bar{z}$ of the series defining the center manifold for a scalar delay differential equation On the computation of the term $w_{21}z^2\bar{z}$ of the series defining the center manifold for a scalar delay differential equation

2011-10-18

arxiv.org/abs/1110.4090v2

Mathematics

Dynamical Systems

Presented at the Conference on Applied and Industrial Mathematics- CAIM 2011, Iasi, Romania, 22-25 September, 2011. Preprint

Scientific paper

In computing the third order terms of the series of powers of the center manifold at an equilibrium point of a scalar delay differential equation, with a single constant delay $r>0,$ some problems occur at the term $w_{21}z^2\bar{z}.$ More precisely, in order to determine the values at 0, respectively $-r$ of the function $w_{21}(\,.\,),$ an algebraic system of equations must be solved. We show that the two equations are dependent, hence the system has an infinity of solutions. Then we show how we can overcome this lack of uniqueness and provide a formula for $w_{21}(0).$

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