#### For citation:

Kurkina E. S., Knyazeva E. N. Sergey P. Kurdyumov and his evolutionary model of dynamics of complex systems. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2013, vol. 21, iss. 4, pp. 135-217. DOI: 10.18500/0869-6632-2013-21-4-135-217

# Sergey P. Kurdyumov and his evolutionary model of dynamics of complex systems

Sergei P. Kurdyumov (1928–2004) and his distinguished contribution in the development of the modern interdisciplinary theory and methodology of study of complex selforganizing systems, i.e. synergetics, is under consideration in the article. The matter of a mathematical model of evolutionary dynamics of complex systems elaborated by him is demonstrated. The nonlinear equation of heat conductivity serves as a basis of the model. Under certain conditions, it describes dynamics of development of structures of different complexity in the blow-up regime. Methods of calculation of two-dimentional structures which are described by automodel solutions are considered; and their classification is given. The automodel problem is a boundary problem aiming to find eigen-values and eigen-functions for a nonlinear equation of elliptical type on a plane. Proceeding from the analysis of the model, a principle of coevolution was formulated by S.P. Kurdyumov. This is the principle of integration of simple structures into a complex one. Three notions of great significance follow from the principle, and namely: the notion of connection of space and time, the notion of complexity and its nature and the notion evolutionary cycles and switching over different regimes as a necessary mechanism of maintenance of «life» of complex structures. Approaches of possible application of this model for understanding of dynamics of complex social, demographic and geopolitical systems are viewed as well.

- Knyazeva EN, Kurdyumov SP. The foundations of synergy. Synergistic worldview. Moscow: URSS; 2010. 238 p. (In Russian).
- Knyazeva EN, Kurdyumov SP. The foundations of synergy. A man who constructs himself and his future. Moscow: URSS; 2011. 260 p. (In Russian).
- Knyazeva EN, Kurdyumov SP. Synergy as a new worldview: Dialogue with I. Prigozhin. Voprosy Filosofii. 1992;12:3–20.
- Knyazeva EN, Kurdyumov SP. Intuition as self-adjustment. Voprosy Filosofii. 1994. No 2. С.110–122.
- Knyazeva EN, Kurdyumov SP. Anthropic principle in synergy. Voprosy Filosofii. 1997. No 3. С. 62–79.
- Knyazeva EN, Kurdyumov SP. Synergetics: Time nonlinearity and coevolution landscapes. Moscow: Kom Kniga; 2007. 272 p. (In Russian).
- Kurdyumov SP. Proprietary functions of combustion of nonlinear medium and design laws of its organization. Modern problems of mathematical physics and computational mathematics. Moscow: Nauka; 1982. 217–243 p.(In Russian).
- Belavin VA, Kapitza SP, Kurdyumov SP. A mathematical model of global demographic processes with regard to the spatial distribution. Comput. Math. Math. Phys. 1998;38(6):849–865.
- Belavin VA, Kurdyumov SP. Blow-up regimes in a demographic system: Scenario of increase in the nonlinearity. Comput. Math. Math. Phys., 40:2 (2000), 227–239.
- Samarskiy AA, Galaktionov VA, Kurdyumov SP, Mikhailov AP. Exacerbated modes in problems for quasi-linear parabolic equations. Moscow: Nauka; 1987. 480 p. (In Russian).
- Regimes with exacerbation: The evolution of the idea. Ed. Malinetskii GG. Moscow: Fizmatlit; 2006. 312 p. (In Russian).
- Yelenin GG, Kurdyumov SP, Samarskii AA. Nonstationary dissipative structures in a nonlinear heat-conducting medium. U.S.S.R. Comput. Math. Math. Phys. 1983;23(2):80–86.
- Kurdyumov SP, Kurkina ES, Malinetskii GG, Samarskii AA. Dissipative structures in an inhomogeneous nonlinear burning medium. Dokl. Akad. Nauk SSSR. 1980;251(3):587–591.
- Dimova SN, Kaschiev MS, Kurdyumov SP. Numerical analysis of the eigenfunctions for the combustion of a nonlinear medium in the radial-symmetric case. U.S.S.R. Comput. Math. Math. Phys. 1989;29(6):61–73.
- Kurkina ES, Kurdyumov SP. SPECTRUM OF DISSIPATIVE STRUCTURES DEVELOPING IN "BLOW-UP" REGIME. Dolladi akademii nauk. 2004;395(6):743–748.
- Kurdyumov SP, Kurkina ES. The spectrum of the eigenfunctions of a self-similar problem for the nonlinear heat equation with a source. Comput. Math. Math. Phys. 2004;44(9):1539–1556.
- Kurdyumov SP, Kurkina ES, Potapov AB, Samarskii AA. The architecture of multidimensional heat structures. Dokl. Akad. Nauk SSSR. 1984;274(5):1071–1075.
- Kurdyumov SP, Kurkina ES, Potapov AB, Samarskii AA. Complex multidimensional structures of the combustion of a nonlinear medium. U.S.S.R. Comput. Math. Math. Phys. 1986;26(4):148–158.
- Potapov AB. Constructs two-dimensional eigenfunctions of a nonlinear environment. Preprint No. 8. Moscow: KIAM USSR. 1986. 26 p. (In Russian).
- Dimova SN, Kaschiev MS, Koleva MG. Investigation of eigenfunctions for combustion of nonlinear medium in polar coordinates with finite elements method. Matem. Mod. 1992;4(3):76–83.
- Kurkina ES. Two-dimensional and three-dimensional thermal structures in a medium with nonlinear thermal conductivity. Computational Math. and Modeling. 2005;16(3):257–278. DOI:10.1007/s10598-005-0023-8.
- Kurkina ES, Kurdyumov SP. THE QUANTUM PROPERTIES OF NONLINEAR DISSIPATIVE MEDIUM. Dokladi akademii nauk. 2004;399(6):741–746.
- Kurkina ES, Nikol’skii IM. Bifurcation analysis of the spectrum of two-dimensional thermal structures evolving with blow-up. Computational Math. and Modeling. 2006;17(4):320–323. DOI:10.1007/s10598-006-0027-z.
- Kurkina ES. Multilink non-linear medium combustion structures. Preprint № 26. Moscow: KIAM . 2006. 25 p. (In Russian).
- Kurkina ES. THE SPECTRUM OF TWO-DIMENSIONAL LOCALIZED STRUCTURES, WHICH DEVELOP IN THE BLOW-UP REGIME. Dynamics of complex systems. 2007;1(1):71–89.
- Kurkina ES, Nikol’skii IM. Stability and localization of unbounded solutions of a nonlinear heat equation in a plane. Computational Math. and Modeling. 2009;20(4):348–366. DOI:10.1007/s10598-009-9042-1.
- Kapitsa SP. Earth's population growth theory. Moscow: MIPT Publ.; 1997. 82 p. (In Russian).
- Kapitsa SP. Essays on the theory of the growth of mankind. Demographic revolution and information society. Moscow: MMVB; 2008. 63 p. (In Russian).
- Malkov AS, Korotaev AV, Khalturina DA. Mathematical model of population growth, economics, technology and education. New in synergy. New reality, new problems, new generation. Vol. 1. Ed. Malinetskii GG. Moscow: Radiotehnika; 2006. 360 p. (In Russian).
- Ivanov OP. Complexity as a category of evolution. Slognie sistemy. 2011;1(4):48–67.
- Rodoman BB. Territorial ranges and networks. Smolensk: Oikumena; 1999. 256 p. (In Russian).
- Rudenko AP. SELF-ORGANIZATION AND SYNERGETRICS. SYNERGETRIC. Vol. 3. Moscow: MSU Press; 2000. 61–99 p. (In Russian).
- Belavin VA, Knyazeva EN, Kurkina EU. Mathematical modeling of global dynamics of the world community. Nonlinearity in modern natural science. Moscow: LIBROKOM; 2009. 384–408 p. (In Russian).
- Kuretova ED, Kurkina ES. Modeling general laws of spatial-temporal evolution growth and historical cycles. Computational Mathematics and Modeling. New York: Springer. 2010;21(2):70–89. DOI:10.1007/s10598-010-9055-9.
- Kurkina ES, Knyazeva EN. Evolution of spatial structures of the world: Mathematical modeling and worldview effects. Evolution: Debating aspects of global evolutionary processes. Moscow: LIBROKOM; 2011. 274–315 p. (In Russian).
- Kurkina ES. Mathematical modeling of the global evolution of the world community. Demographic explosion and collapse of civilization//History and mathematics. Analysis and simulation of global dynamics. Moscow: LIBROKOM; 2010. 230–277 p. (In Russian).
- Knyazev EN, Kurkina ES. Nature of complexity: Methodological consequences of mathematical modeling of evolution of complex structures. Synergistic paradigm. Synergy of innovation complexity. Moscow: Progress-Traditsiya 2011; 443–463 p. (In Russian).
- Kurkina ES. Modeling global spatial-temporal evolution of society: Hyperbolic growth and historical cycles. Extended Abstract in Conference Proceedings of ICNAAM-2011. American Institute of Physics. 2011;1389:1019–1022. DOI:10.1063/1.3637783.

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