Nonautonomous vector fields on $$S^3$$: Simple dynamics and wild embedding of separatrices

We construct new substantive examples of non-autonomous vector fields on 3-dimensional sphere having a simple dynamics but non-trivial topology. The construction is based two ideas: the theory diffeomorpisms with wild separatrix embedding (Pixton, Bonatti-Grines, etc.) and suspension over diffeomorpism (Lerman-Vainshtein). As result, we get periodic, almost periodic or even nonrecurrent which have finite number special integral curves possessing exponential dichotomy $\R$ such that among them there one saddle curve (with an type (3,2)) wildly embedded two-dimensional unstable three-dimensional stable manifold. All other tend, as $t\to \pm \infty,$ to these curves. Also another $k\ge 2$ tamely separatrices forming mildly frames in sense Debrunner-Fox. In case fields, corresponding specific are period field, they for fields.