  This article studies routes to chaos occurring within a resonance wedge for a
3-parametric family of differential equations acting on a 3-sphere. Our
starting point is an autonomous vector field whose flow exhibits an attracting
heteroclinic network made by two 1-dimensional connections and a 2-dimensional
separatrix between two equilibria with different Morse indices. After changing
the parameters, while keeping the 1-dimensional connections unaltered, we
concentrate our study in the case where the 2-dimensional invariant manifolds
of the equilibria do not intersect. We derive the first return map near the
ghost of the attractor and we reduce the analysis of the system to a
2-dimensional map on the cylinder. Complex dynamical features arise from a
discrete-time Bogdanov-Takens singularity, which may be seen as the organizing
center by which one can obtain infinitely many attracting tori, strange
attractors, infinitely many sinks and non-trivial contracting wandering
domains. These dynamical phenomena occur within a structure that we call
resonance wedge. As an application, we may see the "classical" Arnold tongue as
a projection of a resonance wedge. The results are general, extend to other
contexts and lead to a fine-tuning of the theory.
