Influence of Spatial Inhomogeneity on the Formation of Chaotic Modes at the Self-Organization Process


  • Z. M. Liashenko Sumy State University
  • I. A. Lyashenko Sumy State University, Technische Universit¨at Berlin, Institut f¨ur Mechanik, FG Systemdynamik und Reibungsphysik



self-organization, Lorenz system, strange attractor, order parameter, partial differential equations


The Lorentz system of equations, in which gradient terms are taken into account, has been solved numerically. Three fundamentally different modes of evolution are considered. In the first mode, the spatial distribution of the order parameter permanently changes in time, and domains of two types with positive and negative order parameter values are formed. In the second mode, the order parameter distribution is close to the stationary one. Finally, in the third mode, the order parameter is identical over the whole space. The dependences of the average area of domains, their number, and their total area on the time are calculated in the first two cases. In the third case, the contribution of gradient terms completely vanishes, and a classical Lorenz attractor is realized.


H. Haken. Synergetics: An Introduction. Nonequilibrium Phase Transition and Self-Organization in Physics, Chemistry, and Biology (Springer, 1978) [ISBN: 978-3-642-96469-5].

A.V. Voronov, V.M. Petnikova, V.V. Shuvalov. "Magnetodipole" self-organization of charge carriers in high-Tc superconductors and the kinetics of phase transition. J. Exp. Theor. Phys. 93, 1091 (2001).

R. Meucci, F. Salvadori, K.A.M. Al Naimee et al. Attractor selection in a modulated laser and in the Lorenz circuit. Philos. Trans. Royal Soc. A 366, 475 (2008).

S.E. Boulfelfel, A.R. Oganov, S. Leoni. Understanding the nature of "superhard graphite". Sci. Rep. 2, 47 (2012).

L. Yong, Z.X-. Fang, B.Q-. Sheng. Synchronization and control of autocatalytic chemical reaction in continuous stirring tank reactor. Acta Phys. Sin. 57, 4748 (2008).

V.I. Zasimchuk, E.E. Zasimchuk, A.S. Gatsenko. Self-organization in viscous fluids. Metallofiz. Noveish. Tekhnol. 39, 1435 (2017) (in Russian).

H. Haken. Information and Self-Organization. A Macroscopic Approach to Complex Systems (Springer, 2006) [ISBN: 978-3-540-33023-3].

Y. Holovatch, R. Kenna, S. Thurner. Complex systems: physics beyond physics. Eur. J. Phys. 38, 023002 (2017).

E.N. Lorenz. Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 (1963).;2

I.A. Shuda, S.S. Borysov, A.I. Olemskoi. Noise-induced oscillations in non-equilibrium steady state systems. Phys. Scr. 79, 065001 (2009).

A.I. Olemskoi, O.V. Yushchenko, T.I. Zhilenko. Investigation of conditions for a self-organized transition to the bistable regime of quasi-equilibrium condensation and stripping of the surface. Phys. Solid State 53, 845 (2011).

A.I. Olemskoi, A.V. Khomenko. Three-parameter kinetics of phase transition. Zh. ' Eksp. Teor. Fiz. 110, 2144 (1996) (in Russian).

A.I. Olemskoi, A.V. Khomenko, V.P. Koverda. Explosive crystallization of ultradisperse amorphous film. Physica A 284, 79 (2000).

A.I. Olemskoi, A.V. Khomenko, D.O. Kharchenko. Self-organized criticality within fractional Lorenz scheme. Physica A 323, 263 (2003).

I.A. Lyashenko, N.N. Manko. Synergetic model of boundary friction taking into account spatial nonuniformity of stresses, strain, and temperature. Tech. Phys. 59, 1737 (2014).

Ya.A. Lyashenko. Formation of heterogeneous spatial structures in a boundary lubrication layer during friction. J. Appl. Mech. Tech. Phys. 57, 136 (2016).

A.A. Samarskii, A.V. Gulin. Stability of Difference Schemes (Nauka, 1973) (in Russian).

G.E.P. Box, M.E. Muller. A note on the generation of random normal deviates. Ann. Math. Statist. 29, 610 (1958).

J.W. Kantellhardt, S.A. Zschiegner, E. Koscielny-Bunde et al. Multifractal detrended fluctuation analysis of non-stationary time series. Physica A 316, 87 (2002).

O.I. Olemskoi, S.M. Danyl'chenko, V.M. Borysyuk, I.O. Shuda. Multifractal analysis of X-ray patterns of complex condensed media. Metallofiz. Noveish. Tekhnol. 31, 777 (2009).

B.N.J. Persson. On the fractal dimension of rough surfaces. Tribol. Lett. 54, 99 (2014).

E.S.Gadelmawla, M.M.Koura, T.M.A.Maksoud, I.M.Elewa, H.H. Soliman. Roughness parameters. J. Mater. Process. Technol. 123, 133 (2002).

Q. Li, R. Pohrt, V.L. Popov. Adhesive strength of contacts of rough spheres. Front. Mech. Eng. 5, 1 (2019).

Q. Li, R. Pohrt, I.A. Lyashenko, V.L. Popov. Boundary element method for nonadhesive and adhesive contacts of a coated elastic half-space. Proc. Inst. Mech. Eng. J. 234 (1), 73 (2019).

K.L. Johnson. The adhesion of two elastic bodies with slightly wavy surfaces. Int. J. Solids Struct. 32, 423 (1995).

Z.M. Makukha, S.I. Protsenko, L.V. Odnodvorets, I.Yu. Protsenko. Magneto-strain effect in double-layer film systems. J. Nano-Electron. Phys. 4, 02043 (2012).

A.E. Filippov, S.N. Gorb. Methods of the pattern formation in numerical modeling of biological problems. FU Mech. Eng. 17, 217 (2019).

A.I. Dmitriev, A.Yu. Nikonov, W. Osterle, B.Ch. Jim. Verification of Rabinowicz' criterion by direct molecular dynamics modeling. FU Mech. Eng. 17, 207 (2019).

I.A. Lyashenko, V.N. Borysiuk, N.N. Manko. Statistical analysis of self-similar behaviour in the shear induced melting model. Condens. Matter Phys. 17, 23003 (2014).

V. Perekrestov, V. Latyshev, A. Kornyushchenko, Y. Kosminska. Formation, charge transfer, structural and morphological characteristics of ZnO fractal-percolation nanosystems. J. Electron. Mater. 48, 2788 (2019).

V.M. Latyshev, V.I. Perekrestov, A.S. Kornyushchenko, I.V. Zahaiko. Formation of porous zinc nanosystems using direct and reverse flows of DC magnetron sputtering. Funct. Mater. 24, 154 (2017).

A.S. Kornyushchenko, V.V. Natalich, V.I. Perekrestov. Formation of copper porous structures under near-equilibrium chemical vapor deposition. J. Cryst. Growth 442, 68 (2016).

A.D. Pogrebnjak, V.N. Borisyuk, A.A. Bagdasaryan, O.V. Maksakova, E.V. Smirnova. The multifractal investigation of surface microgeometry of (Ti-Hf-Zr-V-Nb)N nitride coatings. J. Nano-Electron Phys. 6, 04018 (2014).



How to Cite

Liashenko, Z. M., & Lyashenko, I. A. (2020). Influence of Spatial Inhomogeneity on the Formation of Chaotic Modes at the Self-Organization Process. Ukrainian Journal of Physics, 65(2), 130.



General physics