Reduced fractal model for quantitative analysis of averaged micromotions in mesoscale: Characterization of blow-like signals Raoul R. Nigmatullin a , Vyacheslav A. Toboev b , Paolo Lino c , Guido Maione c,⇑ a Radioelectronic and Informative Measurements Techniques Department, Kazan National Research Technical University (KNRTU-KAI), 10 Karl Marx str., 420011 Kazan, Tatarstan, Russian Federation b Department of Mathematics, Chuvash State University, Cheboksary, Russian Federation c Department of Electrical and Information Engineering (DEI), Politecnico di Bari, Via E. Orabona, 4, Bari, Italy article info Article history: Received 4 February 2015 Accepted 26 March 2015 abstract It has been shown that many micromotions in the mesoscale region are averaged in accordance with their self-similar (geometrical/dynamical) structure. This distinctive fea- ture helps to reduce a wide set of different micromotions describing relaxation/exchange processes to an averaged collective motion, expressed mathematically in a rather general form. This reduction opens new perspectives in description of different blow-like signals (BLS) in many complex systems. The main characteristic of these signals is a finite duration also when the generalized reduced function is used for their quantitative fitting. As an example, we describe quantitatively available signals that are generated by bronchial asth- matic people, songs by queen bees, and car engine valves operating in the idling regime. We develop a special treatment procedure based on the eigen-coordinates (ECs) method that allows to justify the generalized reduced fractal model (RFM) for description of BLS that can propagate in different complex systems. The obtained describing function is based on the self-similar properties of the different considered micromotions. This kind of cooperative model is proposed here for the first time. In spite of the fact that the nature of the dynamic processes that take place in fractal structure on a mesoscale level is not well understood, the parameters of the RFM fitting function can be used for construction of calibration curves, affected by various external/random factors. Then, the calculated set of the fitting parameters of these calibration curves can characterize BLS of different complex systems affected by those factors. Though the method to construct and analyze the calibration curves goes beyond the scope of this paper, this result could benefit future studies that will employ the developed reduced models in diagnosis, prevention, and control of unpredicted and undesired phenomena of some engineering applications that possibly exhibit such BLS. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction By blow-like signal (BLS) we denote the response of a complex system that provides a signal having finite dura- tion in time or space, starting from zero, achieving maxi- mal/minimal values during a finite time interval, and in the end tending to zero again. Fig. 1(a)–(d) depict typical http://dx.doi.org/10.1016/j.chaos.2015.03.022 0960-0779/Ó 2015 Elsevier Ltd. All rights reserved. Abbreviations: AFR, amplitude–frequency response; BLR, basic linear relationship; BLS, blow-like signal(s); ECs, eigen-coordinates; LLSM, linear least square method; OMA, one-mode approximation; RFM, reduced fractal model. ⇑ Corresponding author. Tel.: +39 080 5963 247; fax: +39 080 5963 410. E-mail address: guido.maione@poliba.it (G. Maione). Chaos, Solitons & Fractals 76 (2015) 166–181 Contents lists available at ScienceDirect Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos