Perturbation of elliptic operators and complex dynamics of parabolic partial differential equations

Let Omega subset of R-N be a smooth bounded domain. Let Lu := Sigma(i,j=i)(N) partial derivative(i)(g(ij)(x)partial derivative(j)u), x is an element of Omega be a second-order strongly elliptic differential operator with smooth symmetric coefficients. Let B denote the Dirichlet or the Neumann boundary operator. We prove the existence of a smooth potential a : <(Omega)over bar> --> R such that all sufficiently small vector fields on RN+1 can be realized on the centre manifold of the semilinear parabolic equation u(t) = Lu + a(x)u + f(x,u,del u), t > 0, x is an element of Omega, Bu = 0, t > 0, x is an element of partial derivative Omega, by an appropriate nonlinearity f : (x, s, w) is an element of <(Omega)over bar> x R x R-N bar right arrow f(x, s, w) is an element of R. For N = 2, n, k is an element of N, we prove the existence of a smooth potential a : <(Omega)over bar> --> R such that all sufficiently small k-jets of vector fields on R-n can be realized on the centre manifold of the semilinear parabolic equation u(t) = Lu + a(x)u + f(x, u) . del u, t > 0, x is an element of Omega Bu = 0, t > 0, x is an element of partial derivative Omega, by an appropriate nonlinearity f : (x, s) is an element of <(Omega)over bar> x R bar right arrow f(x, s) is an element of R-2 (here, '.' denotes the scalar product in R-2).