The "Cocoon" Bifurcations in Three-Dimensonal Systems With Two Fixed Points

We use a geometric approach to explore the dynamical structures of three-dimensional systems with two fixed points. In particular, we study the "cocoon" bifurcation, which is a series of global and local bifurcations that create an infinite number of heteroclinic orbits (which trace out a cocoon-like tangle in space). We introduce a symbolic dynamics to the intersections of a cross-sectional plane with the manifolds of the fixed points. A one-parameter system relevant to the travelling-wave solution of the Kuramoto–Sivashinsky equation is employed to illustrate this approach. The cocoon bifurcation begins with a rapidly converging sequence of heteroclinic bifurcations (principal sequence), each of which generates a pair of heteroclinic orbits and two new pieces of the two-dimensional manifolds. These bifurcations and the manifold structures are also observed with the time delay function of chaotic scattering. At the limit of the principal sequence, a saddle-node bifurcation occurs, followed by a period doubling cascade, which destroys the elliptical orbit and results in a hyperbolic invariant set. Other heteroclinic bifurcations not in the principal sequence are responsible for the folding and eventually the fractal structures of the two-dimensional manifolds. The cocoon bifurcation is completed when all intersections of the two-dimensional manifolds are transverse. The structurally stable system after the cocoon bifurcation contains a hyperbolic invariant set (a horseshoe) imbedded in two fractal sets of the two-dimensional manifolds. All three sets have the same fractal dimension. The cocoon bifurcation is a route to fully developed chaotic scattering. After a prospective effect, called manifold reconnection, the system undergoes a secondary cocoon bifurcation. Then the system becomes a hyperbolic invariant set that is characterized by a full shift of six symbols and is imbedded in two fractal sets of two-dimensional manifolds. Higher order cocoon bifurcations may become overlapped with the homoclinic bifurcation. These results are general and do not depend on the symmetry of the system. For a physical system represented by the dynamical system, it is conjectured that the structurally stable window after the cocoon bifurcation contains the physically most probable state.