Manifold reconnection in chaotic regimes

Two-dimensional nonmonotonic conservative maps are recognized to be of relevance in modeling a number of nonlinear systems, for instance, laser acceleration of charged particles @1–6#, and the nonlinear flow of magnetic field lines in fusion machines such as tokamaks and others @7,8#. As opposed to more traditional monotonic versions, nonmonotonic maps are characterized by frequency curves that are not monotonic functions of the action variable @9#. In laser accelerators, nonmonotonicity arises as a result of the relativistic mass variation of the accelerating particles @5#; in tokamaks, it arises as a result of the geometrical peculiarities of the relevant background magnetic fields. In any case, nonmonotonicity has a strong influence on the types of bifurcations that can occur in the associated nonlinear dynamics. Period doubling cascades of periodic orbits generally precede a transition to chaotic regimes of these orbits, but tangent bifurcations bear no direct relationship to nonintegrability. Indeed, it has been argued that some of the effects preceding a tangent bifurcation even become meaningless in nonintegrable regimes @1#. To analyze the subject, consider the process depicted in the integrable case of Fig. ~1! @2#. Two chains of fixed points undergo a reconnection, starting from the leftmost panel. Before a tangent bifurcation where elliptic fixed points collapse against hyperbolic points, the separatrices defining the upper chain undergo a reconnecting process with those defining the lower chain—this is seen in Fig. 1~b!. It is precisely due to the smoothness caused by integrability that the reconnection can be seen so clearly. This is why reconnection is thought to be of more relevance in integrable cases. In contrast, the process is generally regarded as of little significance in chaotic regimes because in those situations all separatrices—which shall be correctly called stable and unstable manifolds—would be already interlaced with little global response as relevant control parameters are varied. Speaking in more precise terms, effects associated with reconnections are thought to be unobservable when the elliptic fixed points of the reconnecting chains undergo full cascades of period doublings, before any sort of mutual contact of the relevant manifolds takes place. While reconnections, as defined in Ref. @1#, do not occur in fully chaotic regimes, topological rearrangements of stable and unstable manifolds are possible. We shall use the term ‘‘manifold reconnection’’ to describe such processes. What