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

We construct new substantive examples of nonautonomous vector fields on a $3$-dimensional sphere having simple dynamics but nontrivial topology. The construction is based on two ideas: the theory of diffeomorphisms with wild separatrix embedding and the construction of a nonautonomous suspension over a diffeomorphism. As a result, we obtain periodic, almost periodic, or even nonrecurrent vector fields that have a finite number of special integral curves possessing exponential dichotomy on $\mathbb R$ such that among them there is one saddle integral curve (with a $(3,2)$ dichotomy type) with a wildly embedded $2$-dimensional unstable separatrix and a wildly embedded $3$-dimensional stable manifold. All other integral curves tend to these special integral curves as $t\to \pm \infty$. We also construct other vector fields having $k\ge 2$ special saddle integral curves with the tamely embedded $2$-dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner–Fox. In the case of periodic vector fields, the corresponding specific integral curves are periodic with the period of the vector field, and are almost periodic in the case of an almost periodic vector field.