We consider the dynamics of scalar equations u, = uxx +f(x, u) + c(x)α(u), 0«x«l, where α denotes some weighted spatial average and Dirichlet boundary conditions are assumed. Prescribing f, c, α appropriately, it is shown that complicated dynamics can occur. Specifically, linearisations at equilibria can have any number of purely imaginary eigenvalues. Moreover, the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily, up to any finite order. These results are in marked contrast with the case α = 0, where bounded solutions are known to converge to equilibrium.
|Original language||English (US)|
|Number of pages||26|
|Journal||Proceedings of the Royal Society of Edinburgh: Section A Mathematics|
|State||Published - Jan 1 1990|