We demonstrate a new type of basin boundary for typical chaotic dynamical systems. For the case of a two dimensional map, this boundary has the character of the graph of a function that is smooth and differentiable except on a set of fractal dimensions less than one. In spite of the basin boundary being smooth {open_quotes}almost everywhere,{close_quotes} its fractal dimension exceeds one (implying degradation of one{close_quote}s ability to predict the attractor an orbit approaches in the presence of small initial condition uncertainty). We call such a boundary {ital sporadically fractal}. {copyright} {ital 1999} {ital The American Physical Society}
We study fluctuational transitions in a discrete dynamical system that has two co-existing attractors in phase space, separated by a fractal basin boundary which may be either locally-disconnected or locally-connected. It is shown that, in each case, transitions occur via an accessible point on the boundary. The complicated structure of paths inside the locally-disconnected fractal boundary is determined by a hierarchy of homoclinic original saddles. The most probable escape path from a regular attractor to the fractal boundary is found for the each type of boundary using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.
Every function of n inputs can be efficiently computed by a complete network of n processors in such a way that: If no faults occur, no set of size t n /2 of players gets any additional information (other than the function value), Even if Byzantine faults are allowed, no set of size t n /3 can either disrupt the computation or get additional information. Furthermore, the above bounds on t are tight!