We introduce a two-parameter family of partially hyperbolic' skew products (G(a, t)) maps with one dimensional centre direction. In this family, the parameter a models the central dynamics and the parameter t the unfolding of cycles (that occurs for t = 0). The parameter a also measures the central distortion' of the systems: for small a, the distortion of the systems is small and it increases and goes to infinity as a . The family (G(a, t)) displays some of the main characteristic properties of the unfolding of heterodimensional cycles as intermingled homoclinic classes of different indices and secondary bifurcations via collision of hyperbolic homoclinic classes. For a in (0, log2), the dynamics of (G(a, t)) is always non-hyperbolic after the unfolding of the cycle. However, for a > log4 intervals of t-parameters corresponding to hyperbolic dynamics appear and turn into totally prevalent as a (the density of hyperbolic parameters' goes to 1 as a ). The dynamics of the maps G(a, t) is described using a family of iterated function systems modelling the dynamics in the one-dimensional central direction.

In this article we introduce a one-parameter family of skew product (Gt)t ∈ [−ε, ε] maps exhibiting a heterodimensional cycle such that the number of isolated periodic orbits inside it has not super-exponential growth. The dynamics in the central direction of the maps Gt is described by a one-parameter family of system of iterated functions.

We prove that a Hamiltonian star system, defined on a 2d-dimensional symplectic manifold M (d⩾2), is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the 4-dimensional results in Bessa et al. (2010) [5].