In my classes in dynamical systems theory, we were taught how to detect cycles or cyclic behavior in an ODE (be it dampened, sustained or growing) around a fixed point by looking at the eigenvalues for that point (their imaginary component in particular).
But in higher dimensional systems cycles need not have a center, and in practice one often does not even always know the location of a center even if it exists.
Are there any non-local techniques for identifying cyclic behavior in an ODE which would allow me to detect a cycle without a center? Would such techniques similarly allow me to identify strange attractors?
This is a very nontrivial problem in general: to analyze limit cycles in systems of dimension $d\geq 3$. Still there is a lot of research in this direction. A great source of various material is Periodic Motions by Miklos Farkas.