The limited effectiveness of normal forms: a critical review and extension of local bifurcation studies of the Brusselator PDE

摘要: Abstract The detection and unfolding of degenerate local bifuractions provides one very few generally applicable analytical tools for studying complex dynamics in systems arbitrarily high dimension. Using the Brusselator partial differential equations (PDEs) (Prigogine Lefever, 1968) as motivation main example, we critically review this method. We extend correct previous calculations, presenting explicit formulae from which normal forms accurate to third order may be computed, first time carefully compare bifurcations these with those untransformed restricted a center manifold, Galerkin finite difference approximations original PDE. While judicious use symbolic manipulations makes feasible such high-order manifold form show that conclusions drawn them are limited understanding spatio-temporal complexity chaos. As Guckenheimer (1981) argued, method permits proof existence quasi-periodic motions and, under mild genericity assumptions, Sil'nikov chaos (sub-shifts type), but parameter phase space ranges results applied extremely small.