SYSTEMS WITH A HOMOCLINIC CURVE OF MULTIDIMENSIONAL SADDLE-FOCUS TYPE, AND SPIRAL CHAOS

Consider the space of dynamical systems having an isolated equilibrium point of saddle-focus type with a one- or two-dimensional unstable manifold and a trajectory homoclinic at .The following results are proved:Systems with structurally unstable periodic motions are dense in . Systems with a countable set of stable periodic motions are dense in the open subset of comprised of systems whose second saddle parameter is negative. Neither the subset of consisting of systems satisfying 0$ SRC=http://ej.iop.org/images/0025-5734/73/2/A07/tex_sm_2553_img7.gif/> nor any sufficiently small neighborhood of in the space of all dynamical systems contains a system with stable periodic motions in a sufficiently small neighborhood of the contour .