Skip to main content
SpringerLink
Log in

A novel approach to obtain analytical-numerical solutions of nonlinear Lorenz system

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this research we apply an analytic approach to solve the well-known Lorenz system in the non-chaotic regime. The proposed approach is based on modal expansion by infinite series. The analytical-numerical results show that for real initial conditions and under the non-convective regime the modal expansion series reproduce correctly the dynamical behavior of the solution of the Lorenz system. The validity and reliability of the proposed analytical approach with few terms is tested by its application to the convective and non-convective regime with various parameter values. The main advantage is that the obtained solution is global and is presented in analytical form.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbasbandy, S., Shivanian, E.: Exact analytical solution of a nonlinear equation arising in heat transfer. Phys. Lett. A 374(4), 567–574 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Acedo, L., González-Parra, G., Arenas, A.: An exact global solution for the classical SIRS epidemic model. Nonlinear Anal. Real World Appl. 11(3), 1819–1825 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Acedo, L., González-Parra, G., Arenas, A.: Modal series solution for a epidemic model. Phys A 389, 1151–1157 (2010)

    Article  Google Scholar 

  4. Al-sawalha, M.M., Noorani, M.: Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 15(10), 3022–3034 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alomari, A., Noorani, M., Nazar, R., Li, C.: Homotopy analysis method for solving fractional Lorenz system. Commun. Nonlinear Sci. Numer. Simul. 15(7), 1864–1872 (2010)

    Article  MATH  Google Scholar 

  6. Arenas, A.J., González-Parra, G., Jódar, L., Villanueva, R.J.: Piecewise finite series solution of nonlinear initial value differential problem. Appl. Math. Comput. 212(1), 209–215 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Birger, B.: Application of the Lorenz equations for studying thermal convection in the lithosphere. Izvestiya Phys. Solid Earth 47, 541–554 (2011)

    Article  Google Scholar 

  8. van Buuren, S., Hetzler, H., Hinterkausen, M., Seemann, W.: Novel approach to solve the dynamical porous journal bearing problem. Tribol. Int. 46(1), 30–40 (2012)

    Article  Google Scholar 

  9. Chowdhury, M., Hashim, I., Momani, S.: The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system. Chaos, Solitons Fractals 40(4), 1929–1937 (2009)

    Article  MATH  Google Scholar 

  10. Costin, O.: Topological construction of transseries and introduction to generalized Borel summability. AMS CONM SERIES 373, 137 (2005)

    MathSciNet  Google Scholar 

  11. González-Parra, G., Arenas, A., Jódar, L.: Piecewise finite series solutions of seasonal diseases models using multistage Adomian method. Commun. Nonlinear Sci. Numer. Simul. 14, 3967–3977 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hashim, I., Noorani, M., Ahmad, R., Bakar, S., Ismail, E., Zakaria, A.: Accuracy of the Adomian decomposition method applied to the Lorenz system. Chaos, Solitons Fractals 28(5), 1149–1158 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hongli, A.: Numerical pulsrodons of the -dimensional rotating shallow water system. Phys. Lett. A 375(19), 1921–1925 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liao, S.: Beyond Perturbation. Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC (2004)

  15. Lorenz, E.: Deterministic nonperiodic flow1. J. Atmosph. Sci. 20, 130 (1963)

    Article  Google Scholar 

  16. Lu, X., Lu, T., Viljanen, M.: A new analytical method to simulate heat transfer process in buildings. Appl. Therm. Eng. 26(16), 1901–1909 (2006)

    Article  Google Scholar 

  17. Allan, M.F.: Construction of analytic solution to chaotic dynamical systems using the homotopy analysis method. Chaos, Solitons Fractals 39(4), 1744–1752 (2009)

    Article  MATH  Google Scholar 

  18. Mahmud, M., Hashim, I.: Effects of a magnetic field on chaotic convection in fluid layer heated from below. Int. Commun. Heat Mass Transfer 38(4), 481–486 (2011)

    Article  Google Scholar 

  19. Sparrow, C.: The Lorenz equations: Bifurcations, Chaos,and strange attractors. Springer-Verlag (1982)

  20. Tignol, J.P.: Galois theory of algebraic equations. World Scientific, Singapore (2001)

    Book  MATH  Google Scholar 

  21. Zorrilla, S.E., Rubiolo, A.C.: Modelling average concentrations of salt and salt substitute in partial or total volumes of semihard cylindrical cheeses. Int. J. Food Sci. Technol. 33(6), 501–508 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Grupo de Matemática Multidisciplinar (GMM), Fac. de Ingeniería, Universidad de Los Andes, Mérida, Venezuela

    Gilberto González-Parra

  2. Centro de Investigaciones en Matemática Aplicada (CIMA), Universidad de Los Andes, Mérida, Venezuela

    Gilberto González-Parra

  3. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019, USA

    Gilberto González-Parra

  4. Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Valencia, España

    Luis Acedo

  5. Departamento de Matemáticas y Estadística, Universidad de Córdoba, Montería, Colombia

    Abraham J. Arenas

  6. Grupo Teseeo, Universidad del Sinú, Montería, Colombia

    Abraham J. Arenas

Authors
  1. Gilberto González-Parra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luis Acedo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abraham J. Arenas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilberto González-Parra.

Rights and permissions

Reprints and permissions

About this article

Cite this article

González-Parra, G., Acedo, L. & Arenas, A.J. A novel approach to obtain analytical-numerical solutions of nonlinear Lorenz system. Numer Algor 67, 93–107 (2014). https://doi.org/10.1007/s11075-013-9776-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-013-9776-x

Keywords

Search

Navigation