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Self-organized criticality is a concept of avalanche-like phenomena observed in some complex nonlinear systems with scale invariance and arising in the absence of a control factor. The absence of a control factor in the vicinity of criticality leads to uncertainty in the formalism of the causality principle. It is found that, taking into account the violation of the causality principle, rare sets of random numbers are reduced to only two topological patterns similar to two characteristic sections of the electrocardiogram – peak connectivity and relaxation. The achievement of a critical state is determined by stacking data into topological patterns, whereas the distribution of the connectivity index obeys the power law of distribution.
The method of topological patterns is based on a formula for a new physical characteristic of a fractal manifold – the connectivity index, which has no analog in spaces of integer dimension. In the applications in the field of information technology and measuring technology, where there is a problem of choosing an objective quantitative criterion for the noise immunity of devices, the SNR name is used, which is the signal-to-noise ratio. The connectivity index or SNR is invariant with respect to any linear transformations of the original data, whereas for sets of Gauss and Bessel functions, it does not depend on the granulation number, which differs from other formulations of SNR. The article provides a formula for the connectivity index or SNR.
The connectivity index is defined in a fractal manifold, where the representation of a complex nonlinear system with a tendency to self-organization in ordinary space is replaced by a description of a simple linear system in a fractal space. The construction of a fractal manifold becomes possible when the principle of causality is violated. The simulation of inelastic collisions leads to an explosive increase in criticality among sets of random numbers or to the process of detonation – as the creation of order out of chaos.
Keywords:self-organized criticality SOC, fractal manifold, causality principle, connectivity index, signal-to-noise ratio SNR, random number generator, detonation.
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