Achievements investigation of chaotic behavior of a system with delayed feedback, detecting the presence of different kinds of instability and studied influence of delay time on temporal intermittency research in analytical features of the averaging of local functional matrices, allowing improvement of calculation of characteristic exponents research in the method of modeling the dynamics of topological instability for an attractor reconstructed from time series, that provides information about the system (including optimal embedding) development of the validity proved by numerical calculations with both fractal multidimensional attractor sets and real time series (digitized electrocardiogram signal) development of robust algorithms for fractal analysis research in self organization processes in active media, preferably for activator-inhibitor systems