A differential equation is a Hilbert space with all solutions bounded but with so finite nontrivial invariant measure is constructed. In fact, it is shown that all solutions to this equation converge weakly to the origin, nonetheless, there is no stationary point. Moreover, so solution has a non-empty -set.
The existence of the normalizing transformation completely decoupling the stable dynamic from the center manifold dynamic is proved. A numerical procedure for the calculation of the asymptotic series for the decoupling normalizing transformation is proposed. The developed method is especially important for the perturbation theory of center manifold and, in particular, for the local stabilization theory. In the paper some sufficient conditions for local stabilization are given.
We provide existence and stability results for semilinear Dirichlet problems with nonlinearities satisfying some general local growth conditions. We derive a general abstract result which we then apply to prove the existence of solutions, their stability and continuous dependence on parameters for a sixth order ODE with Dirichlet type boundary data.