Systems with a homoclinic curve of multidimensional saddle-focus, and spiral chaos

Consider the space $\mathscr B^1$ of dynamical systems having an isolated equilibrium point $O$ of saddle-focus type with a one- or two-dimensional unstable manifold and a trajectory $\Gamma$ homoclinic at $O$.
The following results are proved:
1. Systems with structurally unstable periodic motions are dense in $\mathscr B^1$.
2. Systems with a countable set of stable periodic motions are dense in the open subset $\mathscr B^1_s$ of $\mathscr B^1$ comprised of systems whose second saddle parameter $\sigma_2$ is negative.
3. Neither the subset $\mathscr B^1_u$ of $\mathscr B^1$ consisting of systems satisfying $\sigma_2>0$ nor any sufficiently small neighborhood of $\mathscr B^1_u$ in the space of all dynamical systems contains a system with stable periodic motions in a sufficiently small neighborhood of the contour $O\cup\Gamma$.