Optical wavelet transform and local scaling properties of fractals

Adaptive strategies for recognition, noise filtering, control, synchronization and targeting of chaos

Combining knowledge of the local variation rates with some information on the long time trends of a dynamical system, we introduce an adaptive recognition technique consisting in a sequence of variable resolution observation intervals at which the geometrical positions are sampled. The sampling times are chosen so that the sequence of observed points forms a regularized set, in the sense that the separation of adjacent points is almost uniform. We show how this adaptive technique is able to recognize the unstable periodic orbits embedded within a chaotic attractor and stabilize anyone of them even in the presence of noise, through small additive corrections to the dynamics. These techniques have been applied to the synchronization of three chaotic systems, assuring secure communication between a message sender and a message receiver; furthermore they provide robust solutions to the problems of targeting of chaos and of filtering the noise out of an experimental chaotic data set. Implementation of adaptive methods to chaotic Lorenz, three and four dimensional Roessler models and Mackey-Glass delayed system are reported.(c) 1997 American Institute of Physics.