Search for Periodic Solutions of Highly Nonlinear Dynamical Systems

  • L. F. Petrov


Numerical-analytical methods for finding periodic solutions of highly nonlinear autonomous and nonautonomous systems of ordinary differential equations are considered. Algorithms for finding initial conditions corresponding to a periodic solution are proposed. The stability of the found periodic solutions is analyzed using corresponding variational systems. The possibility of studying the evolution of periodic solutions in a strange attractor zone and on its boundaries is discussed, and interactive software implementations of the proposed algorithms are described. Numerical examples are given.


highly nonlinear systems of ordinary differential equations periodic solutions stability of periodic solutions strange attractor deterministic chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Loskutov, “Dynamical chaos: Systems of classical mechanics,” Usp. Fiz. Nauk 50 (9), 939–964 (2007).CrossRefGoogle Scholar
  2. 2.
    The Duffing Equation: Nonlinear Oscillators and Their Behavior, Ed. by I. Kovacic and M. J. Brennan (Wiley, Chichester, 2011).Google Scholar
  3. 3.
    P. J. Holmes, “A nonlinear oscillator with a strange attractor,” Philos. Trans. R. Soc. London Ser. A Math. Phys. Sci. 292 (1394), 419–448 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B. I. Kryukov, Forced Oscillations of Essentially Nonlinear Systems (Mashinostroenie, Moscow, 1984) [in Russian].Google Scholar
  5. 5.
    L. F. Petrov, “Nonlinear effects in economic dynamic models,” Nonlinear Anal. 71, 2366–2371 (2009).MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. F. Petrov, Methods for Dynamic Economic Analysis (Infra-M, Moscow, 2010) [in Russian].Google Scholar
  7. 7.
    M. K. Kerimov and E. V. Selimkhanov, “On exact estimates of the convergence rate of Fourier series for functions of one variable in the space L2[−π,π],” Comput. Math. Math. Phys. 56 (5), 717–729 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. Urabe, “Galerkin’s procedure for nonlinear periodic systems,” Arch. Ration. Mech. Anal. 20, 120–152 (1965).CrossRefzbMATHGoogle Scholar
  9. 9.
    A. A. Abramov and L. F. Yukhno, “A numerical method for solving systems of nonlinear equations,” Comput. Math. Math. Phys. 55 (11), 1794–1801 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. A. Garanzha, A. I. Golikov, Yu. G. Evtushenko, and M. X. Nguen, “Parallel implementation of Newton’s method for solving large_scale linear programs,” Comput. Math. Math. Phys. 49 (8), 1303–1317 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yu. A. Chernyaev, “Convergence of the gradient projection method and Newton’s method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set,” Comput. Math. Math. Phys. 56 (10), 1716–1731 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    L. F. Petrov, “Interactive computational search strategy of periodic solutions in essentially nonlinear dynamics,” Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science, Ed. by M. G. Cojocaru (Springer International, Switzerland, 2015), pp. 355–360. doi 10.1007/978-3-319-12307-3_51CrossRefGoogle Scholar
  13. 13.
    A. Yu. Gornov, Computational Techniques for Solving Optimal Control Problems (Nauka, Novosibirsk, 2009) [in Russian].Google Scholar
  14. 14.
    A. B. Dorzhieva and L. F. Petrov, “Numerical study of limit cycles of a dynamical system using the OPTCONA software code,” Proceedings of Lyapunov Conference 2012 (2012), pp. 14.Google Scholar
  15. 15.
    B. P. Demidovich, Lectures on Mathematical Stability Theory (Nauka, Moscow, 1967) [in Russian].zbMATHGoogle Scholar
  16. 16.
    M. J. Feigenbaum, “Universal behavior in nonlinear systems,” Los Alamos Sci. 1, 4–27 (1980).MathSciNetGoogle Scholar
  17. 17.
    A. S. Antipin, “Saddle gradient feedback-controlled processes,” Autom. Remote Control 55 (2), 311–320 (1994).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Plekhanov Russian University of EconomicsMoscowRussia

Personalised recommendations