About Direct Linearization Methods for Nonlinearity

  • Alishir A. AlifovEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1126)


All real dynamic systems are nonlinear and potentially oscillatory, despite their separation into various types (physical, chemical, biological, economic, etc.). And the use of linear models that are valid only for small changes in parameters is associated with mathematical difficulties (finding solutions to nonlinear equations). Known methods of analysis and calculation of nonlinear systems have a significant drawback: high labor intensity and time - consuming. In contrast, direct linearization methods reduce these disadvantages by several orders of magnitude. Below are the methods of direct linearization for the calculation of nonlinear systems. Direct linearization of nonlinearity is considered in two cases. In the first case, the nonlinear function depends on one variable, and in the second - on two. Direct linearization methods are compared with a known averaging method. The procedure for applying direct linearization methods is described for calculating oscillatory systems interacting with energy sources.


Method Nonlinearity Direct linearization Oscillations 


  1. 1.
    Alifov, A.A., Frolov, K.V.: Interaction of Non-Linear Oscillatory Systems with Energy Sources. Hemisphere Publishing Corporation and Taylor & Francis Group, New York (1990)Google Scholar
  2. 2.
    Alifov, A.A.: The Fundamental Principle Which Operates the Universe. RCD, Moscow (2012). (in Russian)Google Scholar
  3. 3.
    Bogolyubov, N.N., Mitropolsky, Y.A.: Asymptotic methods in the theory of nonlinear oscillations. Nauka, Moscow (1974). (in Russian)Google Scholar
  4. 4.
    Vibrations in Technology: A Reference Book in 6 Volumes, vol. 2. Oscillations of Nonlinear Mechanical Systems. Mechanical Engineering, Moscow, Russia (1974). (in Russian)Google Scholar
  5. 5.
    Mitropolsky, Y.A., Van Dao, N.: Applied Asymptotic Methods in Nonlinear Oscillations. Springer, Dordrecht (1997)Google Scholar
  6. 6.
    Hayashi, Ch.: Nonlinear Oscillations in Physical Systems. Princeton University Press, Princeton (2014)Google Scholar
  7. 7.
    Esmailzadeh, E., Younesian, D., Askari, H.: Analytical Methods in Nonlinear Oscillations: Approaches and Applications. Springer, Dordrecht (2019)CrossRefGoogle Scholar
  8. 8.
    He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Modern Phys. B 20(10), 1141–1199 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Wang, Q., Fu, F.: Variational iteration method for solving differential equations with piecewise constant arguments. Int. J. Eng. Manuf. (IJEM) 2(2), 36–43 (2012). Scholar
  11. 11.
    Lin, Y., Zhou, L., Bao, L.: A parameter free iterative method for solving projected generalized Lyapunov equations. Int. J. Eng. Manuf. (IJEM) 2(1), 62 (2012). Scholar
  12. 12.
    Burton, T.D.: A perturbation method for certain nonlinear oscillators. Int. J. Nonl. Mech. 19(5), 397 (1984)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lu, Q.S.: Special issue on advances of nonlinear dynamics, vibrations and control in China. Int. J. Nonl. Sci. Num. Simul. 6(1), preface (2005)Google Scholar
  14. 14.
    Chen, D.-X., Liu, G.-H.: Oscillatory behavior of a class of second-order nonlinear dynamic equations on time scales. Int. J. Eng. Manuf. (IJEM) 1(6) (2011). Scholar
  15. 15.
    Dai, L., Fan, L.: Analytical and numerical approaches to characteristics of linear and nonlinear vibratory systems under piecewise discontinuous disturbances. Comm. Nonl. Sci. Num. Simul. 9, 417–429 (2004)CrossRefGoogle Scholar
  16. 16.
    Panovko, Ya.G.: Fundamentals of Applied Theory of Vibrations and Shock. Mechanical Engineering, Leningrad, Russia (1976). (in Russian)Google Scholar
  17. 17.
    Loytsyansky, L.G.: Free and forced oscillations in the presence of a quadratic and intermediate between the linear and quadratic laws of resistance. Izv. Academy of Sciences of the USSR, Engineering Collection, vol. 18 (1954). (in Russian)Google Scholar
  18. 18.
    Alifov, A.A.: Method of the direct linearization of mixed nonlinearities. J. Mach. Manuf. Reliab. 46(2), 128–131 (2017). Scholar
  19. 19.
    Alifov, A.A., Farzaliev, M.G., Jafarov, E.N.: Dynamics of a self-oscillatory system with an energy source. Russ. Eng. Res. 38(4), 260–262 (2018). Scholar
  20. 20.
    Alifov, A.A.: On the calculation by the method of direct linearization of mixed oscillations in a system with limited power-supply. In: Hu, Z., Petoukhov, S., Dychka, I., He, M. (eds.) Advances in Computer Science for Engineering and Education II, ICCSEEA 2019. Advances in Intelligent Systems and Computing, vol. 938, pp. 23–31. Springer, Cham (2019)Google Scholar
  21. 21.
    Gourary, M.M., Rusakov, S.G.: Analysis of oscillator ensemble with dynamic couplings. In: Hu, Z., Petoukhov, S., He, M. (eds.) Advances in Artificial Systems for Medicine and Education II, AIMEE 2018. Advances in Intelligent Systems and Computing, vol. 902. Springer, Cham (2018)Google Scholar
  22. 22.
    Acebrón, J.A., et al.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77(1), 137–185 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bhansali, P., Roychowdhury, J.: Injection locking analysis and simulation of weakly coupled oscillator networks. In: Li, P., et al. (eds.) Simulation and Verification of Electronic and Biological Systems, pp.71–93. Springer (2011)Google Scholar
  24. 24.
    Ashwin, P., Coombes, S., Nicks, R.J.: Mathematical frameworks for oscillatory network dynamics in neuroscience. J. Math. Neurosci. 6(2), 1–92 (2016)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ziabari, M.T., Sahab, A.R., Fakhari, S.N.S.: Synchronization new 3D chaotic system using brain emotional learning based intelligent controller. Int. J. Inf. Technol. Comput. Sci. (IJITCS) 7(2), 80–87 (2015). Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Mechanical Engineering Research Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations