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Investigation of Lorenz Equation System with Variable Step Size Strategy

Year 2019, Volume: 11, 16 - 20, 30.12.2019

Abstract

In this study, variable step size strategy has been considered to analyze the numerical solution of the Lorenz system with chaotic structure. Phase portraits have been obtained for this chaotic system. The effectiveness of the variable step size strategy for the solution of this chaotic system has been discussed.

References

  • Afacan, E., Yard{\i}m, F.E., {\em Simulation of a communication system using Lorenz-based differential chaos shift keying (DCSK)model}, J. Fac. Eng. Arch. Gazi Univ., \textbf{25(1)}(2010), 101--110.
  • Christodoulou N.S., {\em An algorithm using Runge- Kutta methods of orders 4 and 5 for systems of ODEs}, IJNMA, \textbf{2(1)}(2009), 47--57.
  • \c{C}elik, K., Kurt, E., A new image encryption algorithm basd on Lorenz system, Electronics, Computers and Artificial Intelligence, International Conference-8th edition, 30 June-02 July, Bucharest, Romania, 2016.
  • \c{C}elik K{\i}z{\i}lkan, G., Step size strategies on the numerical integration of the systems of differential equations, Ph.D. Thesis, Selcuk University Graduate Natural and Applied Sciences, Konya, 2009 (in Turkish).
  • \c{C}elik K{\i}z{\i}lkan, G., Ayd{\i}n, K., {\em Step size strategies for the numerical integration of systems differential equations}, J. Comput. Appl. Math., \textbf{236(15)}(2012), 3805--3816.
  • El- Basha, O., El- Shahat, A. Fayed, H.,{\em Chaos theory and Lorenz attractors}, Sohag Journal of Sciences, \textbf{(1)(1)}(2016), 7--12.
  • Guellal, S., Grimalt, P., Cherruault, Y., {\em Numerical Study of Lorenz’ s equation by the Adomian method}, Computers Math. Applic., \textbf{(33)(3)}(1997), 25--29.
  • Guran, A., Ahmadi, G., {\em An Enchanced numerical solution of the Lorenz system by means of the differential Quadrature Method}, AMIM, \textbf{(17)(1)} (2012), 16--30.
  • Hateley, J., The Lorenz System, Lecture Notes, http://web.math.ucsb.edu/~jhateley/paper/lorenz.pdf, (Access date: 23.09.2019).
  • \.{I}nce Polat, S., Evaluation of approximate solutions of the Lorenz system, Master Thesis, The Graduate School of Natural and Aplied Science of Necmettin Erbakan University, Konya, 2019 (in Turkish).
  • Li, J., Wang, Y., Zhang, W., {\em Numerical simulation of the Lorenz-type chaotic system using Barycentric Lagrange interpolation collocation method}, Hindawi Advances in Mathematical Physics, \textbf{(2019)}(2019), 1--10.
  • Lorenz, E.N., {\em Deterministic nonperiodic flow}, Journal of the Atmospheric Sciences, \textbf{20}(1963), 130--141.
  • Pchelintsev, A.N., {\em Numerical and physical modeling of the dynamics of the Lorenz system}, Numerical Analysis and Applications, \textbf{7(2)}(2014), 159--167.
  • Pehlivan, \.{I}., Uyaro\u{g}lu, Y,. {\em A new chaotic attractor from general Lorenz system family and its electronic experimental implementation}, Turk J Elec Eng \& Comp Sci, \textbf{18(2)}(2010), 171--184.
  • Sermutlu, E., {\em Comparison of Runge-Kutta Methods Order 4 and 5 on Lorenz Equations}, Cankaya University Journal of Arts and Sciences, \textbf{1(1)}(2004).
  • Mathematics Libretexts, https://math.libretexts.org/Bookshelves/Applied\_Mathematics/Book\%3A\_Introduction\_to\_ the\_Modeling\_and\_Analysis\_of\_ Complex\_Systems\_ (Sayama)/09\%3A\_Chaos/9.04\%3A\_Chaos\_in\_Continuous-Time\_Model, (Access date: 23.09.2019).
Year 2019, Volume: 11, 16 - 20, 30.12.2019

Abstract

References

  • Afacan, E., Yard{\i}m, F.E., {\em Simulation of a communication system using Lorenz-based differential chaos shift keying (DCSK)model}, J. Fac. Eng. Arch. Gazi Univ., \textbf{25(1)}(2010), 101--110.
  • Christodoulou N.S., {\em An algorithm using Runge- Kutta methods of orders 4 and 5 for systems of ODEs}, IJNMA, \textbf{2(1)}(2009), 47--57.
  • \c{C}elik, K., Kurt, E., A new image encryption algorithm basd on Lorenz system, Electronics, Computers and Artificial Intelligence, International Conference-8th edition, 30 June-02 July, Bucharest, Romania, 2016.
  • \c{C}elik K{\i}z{\i}lkan, G., Step size strategies on the numerical integration of the systems of differential equations, Ph.D. Thesis, Selcuk University Graduate Natural and Applied Sciences, Konya, 2009 (in Turkish).
  • \c{C}elik K{\i}z{\i}lkan, G., Ayd{\i}n, K., {\em Step size strategies for the numerical integration of systems differential equations}, J. Comput. Appl. Math., \textbf{236(15)}(2012), 3805--3816.
  • El- Basha, O., El- Shahat, A. Fayed, H.,{\em Chaos theory and Lorenz attractors}, Sohag Journal of Sciences, \textbf{(1)(1)}(2016), 7--12.
  • Guellal, S., Grimalt, P., Cherruault, Y., {\em Numerical Study of Lorenz’ s equation by the Adomian method}, Computers Math. Applic., \textbf{(33)(3)}(1997), 25--29.
  • Guran, A., Ahmadi, G., {\em An Enchanced numerical solution of the Lorenz system by means of the differential Quadrature Method}, AMIM, \textbf{(17)(1)} (2012), 16--30.
  • Hateley, J., The Lorenz System, Lecture Notes, http://web.math.ucsb.edu/~jhateley/paper/lorenz.pdf, (Access date: 23.09.2019).
  • \.{I}nce Polat, S., Evaluation of approximate solutions of the Lorenz system, Master Thesis, The Graduate School of Natural and Aplied Science of Necmettin Erbakan University, Konya, 2019 (in Turkish).
  • Li, J., Wang, Y., Zhang, W., {\em Numerical simulation of the Lorenz-type chaotic system using Barycentric Lagrange interpolation collocation method}, Hindawi Advances in Mathematical Physics, \textbf{(2019)}(2019), 1--10.
  • Lorenz, E.N., {\em Deterministic nonperiodic flow}, Journal of the Atmospheric Sciences, \textbf{20}(1963), 130--141.
  • Pchelintsev, A.N., {\em Numerical and physical modeling of the dynamics of the Lorenz system}, Numerical Analysis and Applications, \textbf{7(2)}(2014), 159--167.
  • Pehlivan, \.{I}., Uyaro\u{g}lu, Y,. {\em A new chaotic attractor from general Lorenz system family and its electronic experimental implementation}, Turk J Elec Eng \& Comp Sci, \textbf{18(2)}(2010), 171--184.
  • Sermutlu, E., {\em Comparison of Runge-Kutta Methods Order 4 and 5 on Lorenz Equations}, Cankaya University Journal of Arts and Sciences, \textbf{1(1)}(2004).
  • Mathematics Libretexts, https://math.libretexts.org/Bookshelves/Applied\_Mathematics/Book\%3A\_Introduction\_to\_ the\_Modeling\_and\_Analysis\_of\_ Complex\_Systems\_ (Sayama)/09\%3A\_Chaos/9.04\%3A\_Chaos\_in\_Continuous-Time\_Model, (Access date: 23.09.2019).
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Saniye İnce Polat This is me 0000-0003-3832-2235

Gülnur Çelik Kızılkan 0000-0003-1538-082X

Publication Date December 30, 2019
Published in Issue Year 2019 Volume: 11

Cite

APA İnce Polat, S., & Çelik Kızılkan, G. (2019). Investigation of Lorenz Equation System with Variable Step Size Strategy. Turkish Journal of Mathematics and Computer Science, 11, 16-20.
AMA İnce Polat S, Çelik Kızılkan G. Investigation of Lorenz Equation System with Variable Step Size Strategy. TJMCS. December 2019;11:16-20.
Chicago İnce Polat, Saniye, and Gülnur Çelik Kızılkan. “Investigation of Lorenz Equation System With Variable Step Size Strategy”. Turkish Journal of Mathematics and Computer Science 11, December (December 2019): 16-20.
EndNote İnce Polat S, Çelik Kızılkan G (December 1, 2019) Investigation of Lorenz Equation System with Variable Step Size Strategy. Turkish Journal of Mathematics and Computer Science 11 16–20.
IEEE S. İnce Polat and G. Çelik Kızılkan, “Investigation of Lorenz Equation System with Variable Step Size Strategy”, TJMCS, vol. 11, pp. 16–20, 2019.
ISNAD İnce Polat, Saniye - Çelik Kızılkan, Gülnur. “Investigation of Lorenz Equation System With Variable Step Size Strategy”. Turkish Journal of Mathematics and Computer Science 11 (December 2019), 16-20.
JAMA İnce Polat S, Çelik Kızılkan G. Investigation of Lorenz Equation System with Variable Step Size Strategy. TJMCS. 2019;11:16–20.
MLA İnce Polat, Saniye and Gülnur Çelik Kızılkan. “Investigation of Lorenz Equation System With Variable Step Size Strategy”. Turkish Journal of Mathematics and Computer Science, vol. 11, 2019, pp. 16-20.
Vancouver İnce Polat S, Çelik Kızılkan G. Investigation of Lorenz Equation System with Variable Step Size Strategy. TJMCS. 2019;11:16-20.