ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
Процессы Управления

A Numerical-analytical Method for Constructing Periodic Solutions of the Lorenz System


Alexander N. Pchelintsev

Tambov State Technical University


This article describes a method for constructing approximations to periodic solutions of dynamic Lorenz system with classical values of the system parameters. The author obtained a system of nonlinear algebraic equations in general form concerning of the cyclic frequency, constant terms and amplitudes of harmonics that make up harmonic approximations to the desired solutions. The initial approximation for the Newton method is selected, which converges to a solution describing a periodic solution different from the equilibrium position. The results of a computational experiment are presented. The results are verified using high-precision calculations.

Ключевые слова


  1. Lorenz, E. N. Deterministic Nonperiodic Flow, Journal of the Atmospheric Sciences, vol. 20, no. 2 (1963), pp. 130-141
  2. Tucker, W. A Rigorous ODE Solver and Smale's 14th Problem, Foundations of Computational Mathematics, vol. 2, no. 1 (2002), pp. 53-117
  3. Rabinovich, M. I. Stochastic Self-Oscillations and Turbulence, Soviet Physics Uspekhi, vol. 21, no. 5 (1978), pp. 443-469
  4. Galias, Z., Tucker, W. Validated Study of the Existence of Short Cycles for Chaotic Systems Using Symbolic Dynamics and Interval Tools, International Journal of Bifurcation and Chaos, vol. 21, no. 2 (2011), pp. 551-563
  5. Lozi, R. Can We Trust in Numerical Computations of Chaotic Solutions of Dynamical Systems?, Topology and Dynamics of Chaos. In Celebration of Robert Gilmore's 70th Birthday. - World Scientific Series in Nonlinear Science Series A, vol. 84 (2013), pp. 63-98
  6. Viswanath, D. The Fractal Property of the Lorenz Attractor, Physica D: Nonlinear Phenomena, vol. 190, no. 1-2 (2004), pp. 115-128
  7. Viswanath, D. The Lindstedt-Poincare Technique as an Algorithm for Computing Periodic Orbits, SIAM Review, vol. 43, no. 3 (2001), pp. 478-495
  8. Pchelintsev, A. N. Numerical and Physical Modeling of the Dynamics of the Lorenz System, Numerical Analysis and Applications, vol. 7, no. 2 (2014), pp. 159-167
  9. Neymeyr, K., Seelig, F. Determination of Unstable Limit Cycles in Chaotic Systems by Method of Unrestricted Harmonic Balance, Zeitschrift fü r Naturforschung A, vol. 46, no. 6 (1991), pp. 499-502
  10. Luo, A. C. J., Huang, J. Approximate Solutions of Periodic Motions in Nonlinear Systems via a Generalized Harmonic Balance, Journal of Vibration and Control, vol. 18, no. 11 (2011), pp. 1661-1674
  11. Luo, A. C. J. Toward Analytical Chaos in Nonlinear Systems, John Wiley & Sons, Chichester, ISBN: 978-1-118-65861-1, 2014, 258 pp
  12. Luo, A. C. J., Guo, S. Analytical Solutions of Period-1 to Period-2 Motions in a Periodically Diffused Brusselator, Journal of Computational and Nonlinear Dynamics, vol. 13, no. 9, 090912 (2018), 8 pp
  13. Pchelintsev, A. N. The Programs for Finding of Periodic Solutions in the Lorenz Attractor, GitHub,
  14. Tolstov, G. P. Fourier Series, Dover Publications, New York (1962), 336 pp

Полный текст (pdf)