Chaotic Clustering: Fragmentary Synchronization of Fractal Waves

Chaotic neural networks seized attention of scientists from various points of view due to the amazing effects they produce. Phenomenology of structure formation in nature inspired scholars to mimic complex and with the same time quasi-optimal solutions to generate artificial systems with similar capabilities. One of the dominant ways to provide collective dynamics of previously unordered elements is self-synchronization that happens without any outside enforcement. In theory of dynamic chaos and chaotic synchronization were discovered different types of synchronous behaviour of coupled dynamic systems: complete, lag, generalized, phase, time-scale, frequency, partial synchronizations (Pikovsky & Maistrenko, 2008; Anishchenko & et. al., 2007; Koronovskii & Maistrenko, 2009; Fujisaka & Shimada, 1997, Kapitaniak & Maistrenko 1996; Kurths & et. al., 1997). Most of the results were obtained by computer modelling and visualization techniques. Unpredictable and instable long time period behaviour stems from peculiarities of elements self-dynamics governed by deterministic function that predetermine chaotic behaviour (Lyapunov coefficients are positive). Research subject in numerous articles is collective dynamics of chaotic elements that somehow happens to be stable in terms of group oscillations, but vulnerable when individual trajectory is considered (Politi & Torcini, 2010; Liu & et. al., 2010 ; Pikovsky & et. al., 2003; Manrubia & Mikhailov, 1999). Strange attractors correspond to synchronous clusters (Benderskaya & Zhukova, 2008; Chen & et. al., 2006; Anishenko & et. al., 2002). Extreme complexity of elements individual dynamics constrains the type of interconnection strength to be homogeneous in overwhelming majority of articles. Most of them consider dependence of synchronous regimes from strength of coupling equal for all elements. This limitation prevents from further generalization of results and makes them to be partial solutions. In most recent researches are considered heterogeneous couplings strength. Heterogeneous field of links between elements extends greatly the analysis complexity (especially mathematical analysis) but allows to reveal interesting effects (Inoue & Kaneko, 2010; Li & et. al., 2004; Popovych & et. al., 2000). In this paper we consider the analysis of fragmentary synchronization of fractal waves generated by large dimension inhomogeneous chaotic neural network. In our previous papers fragmentary synchronization phenomenon was discovered and applied. Further research of new synchronization type shows that we can speak about fractal structure of