  This article concerns arbitrary finite heteroclinic networks in any phase
space dimension whose vertices can be a random mixture of equilibria and
periodic orbits. In addition, tangencies in the intersection of un/stable
manifolds are allowed. The main result is a reduction to algebraic equations of
the problem to find all solutions that are close to the heteroclinic network
for all time, and their parameter values. A leading order expansion is given in
terms of the time spent near vertices and, if applicable, the location on the
non-trivial tangent directions. The only difference between a periodic orbit
and an equilibrium is that the time parameter is discrete for a periodic orbit.
The essential assumptions are hyperbolicity of the vertices and transversality
of parameters. Using the result, conjugacy to shift dynamics for a generic
homoclinic orbit to a periodic orbit is proven. Finally,
equilibrium-to-periodic orbit heteroclinic cycles of various types are
considered.
