The methods for calculation of recurrence plots in space with a scalar product were developed. They enable examining the properties and features of the vector of states of dynamical systems of various complexity in natural and social spheres. A new scientific result is the development of scientific-methodical apparatus to calculate recurrence plots of vectors of states of systems in metric spaces expanded on the basis of scalar product. Two methods for calculation of recurrence plots for vectors of states of complex dynamic systems, which are highly informative, moderately complex and universal in dimensionality of the studied space of states, were proposed. In practice, the proposed methods can be used to calculate and compare recurrence plots of states of the studied systems in metric spaces of different dimensionality without additional normalizing. The functionality of the proposed methods was verified based on experimental observations of concentrations of formaldehyde, ammonia and carbon monoxide in the atmosphere of an industrial city. It was established that at the values of angular size of the area of 10° and 30°, the proposed method for calculation of recurrence plots is more informative, less complex and invariant relative to dimensionality of the space of states. It was shown that the methods for calculation of recurrence plots in space with scalar product make it possible to use them if there are short-time intervals of the absence of observations. It was experimentally determined that in some cases of parameters, the results of computation of recurrence plots based on the developed methods coincide with the results obtained when using the known methods. This indicates a more general nature of the proposed methods.

Pospelov, B., Rybka, E., Meleshchenko, R., Gornostal, S., Shcherbak, S. (2017). Results of experimental research into correlations between hazardous factors of ignition of materials in premises. Eastern-European Journal of Enterprise Technologies, 6 (10 (90)), 50–56. doi: https://doi.org/10.15587/1729-4061.2017.117789

Manuca, R., Savit, R. (1996). Stationarity and nonstationarity in time series analysis. Physica D: Nonlinear Phenomena, 99 (2-3), 134–161. doi: https://doi.org/10.1016/s0167-2789(96)00139-x

Webber, C., Marwan, N. (Eds.) (2015). Recurrence quantification analysis. Understanding Complex Systems. Springer. doi: https://doi.org/10.1007/978-3-319-07155-8

Marwan, N., Webber, C. L., Macau, E. E. N., Viana, R. L. (2018). Introduction to focus issue: Recurrence quantification analysis for understanding complex systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (8), 085601. doi: https://doi.org/10.1063/1.5050929

Recurrence plots and cross recurrence plots. Available at: http://www.recurrence-plot.tk

Pospelov, B., Andronov, V., Rybka, E., Meleshchenko, R., Gornostal, S. (2018). Analysis of correlation dimensionality of the state of a gas medium at early ignition of materials. Eastern-European Journal of Enterprise Technologies, 5 (10 (95)), 25–30. doi: https://doi.org/10.15587/1729-4061.2018.142995

Takens, F. (1981). Detecting strange attractors in turbulence. Lecture Notes in Mathematics, 366–381. doi: https://doi.org/10.1007/bfb0091924

Pospelov, B., Andronov, V., Rybka, E., Popov, V., Semkiv, O. (2018). Development of the method of frequency­temporal representation of fluctuations of gaseous medium parameters at fire. Eastern-European Journal of Enterprise Technologies, 2 (10 (92)), 44–49. doi: https://doi.org/10.15587/1729-4061.2018.125926

Adeniji, A. E., Olusola, O. I., Njah, A. N. (2018). Comparative study of chaotic features in hourly wind speed using recurrence quantification analysis. AIP Advances, 8 (2), 025102. doi: https://doi.org/10.1063/1.4998674

Wendi, D., Marwan, N., Merz, B. (2018). In Search of Determinism-Sensitive Region to Avoid Artefacts in Recurrence Plots. International Journal of Bifurcation and Chaos, 28 (01), 1850007. doi: https://doi.org/10.1142/s0218127418500074

Donner, R. V., Balasis, G., Stolbova, V., Georgiou, M., Wiedermann, M., Kurths, J. (2019). Recurrence‐Based Quantification of Dynamical Complexity in the Earth's Magnetosphere at Geospace Storm Timescales. Journal of Geophysical Research: Space Physics, 124 (1), 90–108. doi: https://doi.org/10.1029/2018ja025318

Garcia-Ceja, E., Uddin, M. Z., Torresen, J. (2018). Classification of Recurrence Plots’ Distance Matrices with a Convolutional Neural Network for Activity Recognition. Procedia Computer Science, 130, 157–163. doi: https://doi.org/10.1016/j.procs.2018.04.025

Neves, F. M., Viana, R. L., Pie, M. R. (2017). Recurrence analysis of ant activity patterns. PLOS ONE, 12 (10), e0185968. doi: https://doi.org/10.1371/journal.pone.0185968

Ozken, I., Eroglu, D., Breitenbach, S. F. M., Marwan, N., Tan, L., Tirnakli, U., Kurths, J. (2018). Recurrence plot analysis of irregularly sampled data. Physical Review E, 98 (5). doi: https://doi.org/10.1103/physreve.98.052215

Eroglu, D., Marwan, N., Stebich, M., Kurths, J. (2018). Multiplex recurrence networks. Physical Review E, 97 (1). doi: https://doi.org/10.1103/physreve.97.012312

Oberst, S., Niven, R., Ord, A., Hobbs, B., Lester, D. (2017). Application of recurrence plots to orebody exploration data. Conference: Target. At University Club, University of Western Australia.

Webber, C. L., Ioana, C., Marwan, N. (Eds.) (2016). Recurrence Plots and Their Quantifications: Expanding Horizons. Springer Proceedings in Physics. doi: https://doi.org/10.1007/978-3-319-29922-8

Birleanu, F.-M., Candel, I., Ioana, C., Gervaise, C., Serbanescu, A., Serban, G. (2012). A vector approach to transient signal processing. 2012 11th International Conference on Information Science, Signal Processing and Their Applications (ISSPA). doi: https://doi.org/10.1109/isspa.2012.6310462

Ioana, C., Digulescu, A., Serbanescu, A., Candel, I., Birleanu, F.-M. (2014). Recent Advances in Non-stationary Signal Processing Based on the Concept of Recurrence Plot Analysis. Translational Recurrences, 75–93. doi: https://doi.org/10.1007/978-3-319-09531-8_5

Pospelov, B., Andronov, V., Rybka, E., Meleshchenko, R., Borodych, P. (2018). Studying the recurrent diagrams of carbon monoxide concentration at early ignitions in premises. Eastern-European Journal of Enterprise Technologies, 3 (9 (93)), 34–40. doi: https://doi.org/10.15587/1729-4061.2018.133127

Pospelov, B., Andronov, V., Rybka, E., Skliarov, S. (2017). Design of fire detectors capable of self-adjusting by ignition. Eastern-European Journal of Enterprise Technologies, 4 (9 (88)), 53–59. doi: https://doi.org/10.15587/1729-4061.2017.108448

Pospelov, B., Andronov, V., Rybka, E., Skliarov, S. (2017). Research into dynamics of setting the threshold and a probability of ignition detection by self­adjusting fire detectors. Eastern-European Journal of Enterprise Technologies, 5 (9 (89)), 43–48. doi: https://doi.org/10.15587/1729-4061.2017.110092

Marwan, N., Carmenromano, M., Thiel, M., Kurths, J. (2007). Recurrence plots for the analysis of complex systems. Physics Reports, 438 (5-6), 237–329. doi: https://doi.org/10.1016/j.physrep.2006.11.001

Mindlin, G. M., Gilmore, R. (1992). Topological analysis and synthesis of chaotic time series. Physica D: Nonlinear Phenomena, 58 (1-4), 229–242. doi: https://doi.org/10.1016/0167-2789(92)90111-y

Zbilut, J. P., Zaldivar-Comenges, J.-M., Strozzi, F. (2002). Recurrence quantification based Liapunov exponents for monitoring divergence in experimental data. Physics Letters A, 297(3-4), 173–181. doi: https://doi.org/10.1016/s0375-9601(02)00436-x

Thiel, M., Romano, M. C., Kurths, J., Meucci, R., Allaria, E., Arecchi, F. T. (2002). Influence of observational noise on the recurrence quantification analysis. Physica D: Nonlinear Phenomena, 171 (3), 138–152. doi: https://doi.org/10.1016/s0167-2789(02)00586-9

Kondratenko, O. M., Vambol, S. O., Strokov, O. P., Avramenko, A. M. (2015). Mathematical model of the efficiency of diesel particulate matter filter. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, 6, 55–61.

Vasiliev, M. I., Movchan, I. O., Koval, O. M. (2014). Diminishing of ecological risk via optimization of fire-extinguishing system projects in timber-yards. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, 5, 106–113.

Dubinin, D., Korytchenko, K., Lisnyak, A., Hrytsyna, I., Trigub, V. (2017). Numerical simulation of the creation of a fire fighting barrier using an explosion of a combustible charge. Eastern-European Journal of Enterprise Technologies, 6 (10 (90)), 11–16. doi: https://doi.org/10.15587/1729-4061.2017.114504

Semko, A., Rusanova, O., Kazak, O., Beskrovnaya, M., Vinogradov, S., Gricina, I. (2015). The use of pulsed high-speed liquid jet for putting out gas blow-out. The International Journal of Multiphysics, 9 (1), 9–20. doi: https://doi.org/10.1260/1750-9548.9.1.9

Kustov, M. V., Kalugin, V. D., Tutunik, V. V., Tarakhno, E. V. (2019). Physicochemical principles of the technology of modified pyrotechnic compositions to reduce the chemical pollution of the atmosphere. Voprosy Khimii i Khimicheskoi Tekhnologii, 1, 92–99. doi: https://doi.org/10.32434/0321-4095-2019-122-1-92-99

Vasyukov, A., Loboichenko, V., Bushtec, S. (2016). Identification of bottled natural waters by using direct conductometry. Ecology, Environment and Conservation, 22 (3), 1171–1176.

Pospelov, B., Rybka, E., Meleshchenko, R., Borodych, P., Gornostal, S. (2019). Development of the method for rapid detection of hazardous atmospheric pollution of cities with the help of recurrence measures. Eastern-European Journal of Enterprise Technologies, 1 (10 (97)), 29–35. doi: https://doi.org/10.15587/1729-4061.2019.155027

Results of experimental research into correlations between hazardous factors of ignition of materials in premises / Pospelov B., Rybka E., Meleshchenko R., Gornostal S., Shcherbak S. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 6, Issue 10 (90). P. 50–56. doi: https://doi.org/10.15587/1729-4061.2017.117789 

Manuca R., Savit R. Stationarity and nonstationarity in time series analysis // Physica D: Nonlinear Phenomena. 1996. Vol. 99, Issue 2-3. P. 134–161. doi: https://doi.org/10.1016/s0167-2789(96)00139-x 

Recurrence quantification analysis / C. Webber, N. Marwan (Eds.) // Understanding Complex Systems. Springer, 2015. doi: https://doi.org/10.1007/978-3-319-07155-8 

Introduction to focus issue: Recurrence quantification analysis for understanding complex systems / Marwan N., Webber C. L., Macau E. E. N., Viana R. L. // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2018. Vol. 28, Issue 8. P. 085601. doi: https://doi.org/10.1063/1.5050929 

Recurrence plots and cross recurrence plots. URL: http://www.recurrence-plot.tk

Analysis of correlation dimensionality of the state of a gas medium at early ignition of materials / Pospelov B., Andronov V., Rybka E., Meleshchenko R., Gornostal S. // Eastern-European Journal of Enterprise Technologies. 2018. Vol. 5, Issue 10 (95). P. 25–30. doi: https://doi.org/10.15587/1729-4061.2018.142995 

Takens F. Detecting strange attractors in turbulence // Lecture Notes in Mathematics. 1981. P. 366–381. doi: https://doi.org/10.1007/bfb0091924 

Development of the method of frequency­temporal representation of fluctuations of gaseous medium parameters at fire / Pospelov B., Andronov V., Rybka E., Popov V., Semkiv O. // Eastern-European Journal of Enterprise Technologies. 2018. Vol. 2, Issue 10 (92). P. 44–49. doi: https://doi.org/10.15587/1729-4061.2018.125926 

Adeniji A. E., Olusola O. I., Njah A. N. Comparative study of chaotic features in hourly wind speed using recurrence quantification analysis // AIP Advances. 2018. Vol. 8, Issue 2. P. 025102. doi: https://doi.org/10.1063/1.4998674 

Wendi D., Marwan N., Merz B. In Search of Determinism-Sensitive Region to Avoid Artefacts in Recurrence Plots // International Journal of Bifurcation and Chaos. 2018. Vol. 28, Issue 01. P. 1850007. doi: https://doi.org/10.1142/s0218127418500074 

Recurrence‐Based Quantification of Dynamical Complexity in the Earth's Magnetosphere at Geospace Storm Timescales / Donner R. V., Balasis G., Stolbova V., Georgiou M., Wiedermann M., Kurths J. // Journal of Geophysical Research: Space Physics. 2019. Vol. 124, Issue 1. P. 90–108. doi: https://doi.org/10.1029/2018ja025318 

Garcia-Ceja E., Uddin M. Z., Torresen J. Classification of Recurrence Plots’ Distance Matrices with a Convolutional Neural Network for Activity Recognition // Procedia Computer Science. 2018. Vol. 130. P. 157–163. doi: https://doi.org/10.1016/j.procs.2018.04.025 

Neves F. M., Viana R. L., Pie M. R. Recurrence analysis of ant activity patterns // PLOS ONE. 2017. Vol. 12, Issue 10. P. e0185968. doi: https://doi.org/10.1371/journal.pone.0185968 

Recurrence plot analysis of irregularly sampled data / Ozken I., Eroglu D., Breitenbach S. F. M., Marwan N., Tan L., Tirnakli U., Kurths J. // Physical Review E. 2018. Vol. 98, Issue 5. doi: https://doi.org/10.1103/physreve.98.052215 

Multiplex recurrence networks / Eroglu D., Marwan N., Stebich M., Kurths J. // Physical Review E. 2018. Vol. 97, Issue 1. doi: https://doi.org/10.1103/physreve.97.012312 

Application of recurrence plots to orebody exploration data / Oberst S., Niven R., Ord A., Hobbs B., Lester D. // Conference: Target. At University Club, University of Western Australia. 2017.

Recurrence Plots and Their Quantifications: Expanding Horizons / C. L. Webber, C. Ioana, N. Marwan (Eds.) // Springer Proceedings in Physics. 2016. doi: https://doi.org/10.1007/978-3-319-29922-8 

A vector approach to transient signal processing / Birleanu F.-M., Candel I., Ioana C., Gervaise C., Serbanescu A., Serban G. // 2012 11th International Conference on Information Science, Signal Processing and their Applications (ISSPA). 2012. doi: https://doi.org/10.1109/isspa.2012.6310462 

Recent Advances in Non-stationary Signal Processing Based on the Concept of Recurrence Plot Analysis / Ioana C., Digulescu A., Serbanescu A., Candel I., Birleanu F.-M. // Springer Proceedings in Mathematics & Statistics. 2014. P. 75–93. doi: https://doi.org/10.1007/978-3-319-09531-8_5 

Studying the recurrent diagrams of carbon monoxide concentration at early ignitions in premises / Pospelov B., Andronov V., Rybka E., Meleshchenko R., Borodych P. // Eastern-European Journal of Enterprise Technologies. 2018. Vol. 3, Issue 9 (93). P. 34–40. doi: https://doi.org/10.15587/1729-4061.2018.133127 

Design of fire detectors capable of self-adjusting by ignition / Pospelov B., Andronov V., Rybka E., Skliarov S. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 4, Issue 9 (88). P. 53–59. doi: https://doi.org/10.15587/1729-4061.2017.108448 

Research into dynamics of setting the threshold and a probability of ignition detection by self­adjusting fire detectors / Pospelov B., Andronov V., Rybka E., Skliarov S. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 5, Issue 9 (89). P. 43–48. doi: https://doi.org/10.15587/1729-4061.2017.110092 

Recurrence plots for the analysis of complex systems / Marwan N., Carmenromano M., Thiel M., Kurths J. // Physics Reports. 2007. Vol. 438, Issue 5-6. P. 237–329. doi: https://doi.org/10.1016/j.physrep.2006.11.001 

Mindlin G. M., Gilmore R. Topological analysis and synthesis of chaotic time series // Physica D: Nonlinear Phenomena. 1992. Vol. 58, Issue 1-4. P. 229–242. doi: https://doi.org/10.1016/0167-2789(92)90111-y 

Zbilut J. P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based Liapunov exponents for monitoring divergence in experimental data // Physics Letters A. 2002. Vol. 297, Issue 3-4. P. 173–181. doi: https://doi.org/10.1016/s0375-9601(02)00436-x 

Influence of observational noise on the recurrence quantification analysis / Thiel M., Romano M. C., Kurths J., Meucci R., Allaria E., Arecchi F. T. // Physica D: Nonlinear Phenomena. 2002. Vol. 171, Issue 3. P. 138–152. doi: https://doi.org/10.1016/s0167-2789(02)00586-9 

Mathematical model of the efficiency of diesel particulate matter filter / Kondratenko O. M., Vambol S. O., Strokov O. P., Avramenko A. M. // Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 2015. Issue 6. P. 55–61.

Vasiliev M. I., Movchan I. O., Koval O. M. Diminishing of ecological risk via optimization of fire-extinguishing system projects in timber-yards // Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu. 2014. Issue 5. P. 106–113.

Numerical simulation of the creation of a fire fighting barrier using an explosion of a combustible charge / Dubinin D., Korytchenko K., Lisnyak A., Hrytsyna I., Trigub V. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 6, Issue 10 (90). P. 11–16. doi: https://doi.org/10.15587/1729-4061.2017.114504 

The use of pulsed high-speed liquid jet for putting out gas blow-out / Semko A., Rusanova O., Kazak O., Beskrovnaya M., Vinogradov S., Gricina I. // The International Journal of Multiphysics. 2015. Vol. 9, Issue 1. P. 9–20. doi: https://doi.org/10.1260/1750-9548.9.1.9 

Physicochemical principles of the technology of modified pyrotechnic compositions to reduce the chemical pollution of the atmosphere / Kustov M. V., Kalugin V. D., Tutunik V. V., Tarakhno E. V. // Voprosy Khimii i Khimicheskoi Tekhnologii. 2019. Issue 1. P. 92–99. doi: https://doi.org/10.32434/0321-4095-2019-122-1-92-99 

Vasyukov A., Loboichenko V., Bushtec S. Identification of bottled natural waters by using direct conductometry // Ecology, Environment and Conservation. 2016. Vol. 22, Issue 3. P. 1171–1176.

Development of the method for rapid detection of hazardous atmospheric pollution of cities with the help of recurrence measures / Pospelov B., Rybka E., Meleshchenko R., Borodych P., Gornostal S. // Eastern-European Journal of Enterprise Technologies. 2019. Vol. 1, Issue 10 (97). P. 29–35. doi: https://doi.org/10.15587/1729-4061.2019.155027