Simple bifurcations leading to hyperbolic attractors

We prove the existence of a new stability boundary of periodic orbits in a high-dimensional case, thereby resolving the problem on a “blue sky catastrophe” in a general one-parameter family. We additionally establish the existence of a codimension-one boundary which separates the Morse-Smale systems from systems with hyperbolic Smale-Williams-like attractors. The route across this boundary is accomplished by the disappearance of a saddle-node periodic orbit. We also study the principal bifurcations of a torus breakdown which lead to Anosov attractors and to multidimensional solenoids.