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We construct new substantive examples of non-autonomous vector fields on3-dimensional sphere having a simple dynamics but non-trivial topology. Theconstruction is based on two ideas: the theory of diffeomorpisms with wildseparatrix embedding (Pixton, Bonatti-Grines, etc.) and the construction of anon-autonomous suspension over a diffeomorpism (Lerman-Vainshtein). As aresult, we get periodic, almost periodic or even nonrecurrent vector fieldswhich have a finite number of special integral curves possessing exponentialdichotomy on $\R$ such that among them there is one saddle integral curve (withan exponential dichotomy of the type (3,2)) having wildly embeddedtwo-dimensional unstable separatrix and wildly embedded three-dimensionalstable manifold. All other integral curves tend, as $t\to \pm \infty,$ to thesespecial integral curves. Also we construct another vector fields having $k\ge2$ special saddle integral curves with tamely embedded two-dimensional unstableseparatrices forming mildly wild frames in the sense of Debrunner-Fox. In thecase of periodic vector fields, corresponding specific integral curves areperiodic with the period of the vector field, and they are almost periodic forthe case of almost periodic vector fields.

Non-autonomous vector fields on $S^3$: simple dynamics and wild separatrices embedding