DOI QR코드

DOI QR Code

Comparison Analysis of Behavior between Differential Equation and Fractional Differential Equation in the Van der Pol Equation

Van der Pol 발진기에서의 미분방정식과 Fractional 미분방정식의 거동 비교 해석

  • Bae, Young-Chul (Division of Electrical.Electronics Communication and Computer Engineering, Chonnam National University)
  • 배영철 (전남대학교 전기.전자통신.컴퓨터공학부)
  • Received : 2015.12.23
  • Accepted : 2016.01.24
  • Published : 2016.01.30

Abstract

Three hundred years ago, the fractional differential equation that is one of concept of fractional calculus released. Now, many researchers continue to try best effort applying into the control engineering, mathematics and physics. In this paper, the dynamics equation which is represented by Van der Pol, represent integer order and fractional order that having real order. Then this paper performs the comparisons between integer and real order as time series and phase portrait according to variation of parameter value for real order.

References

  1. J. J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, New Jersey: Prentice-Hall, 1991.
  2. J. K. Hale, Oscillations in Nonlinear Systems. New York: McGraw-Hill, 1963.
  3. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer, 1983.
  4. Y. Bae, J. Kim, Y. Kim, and Y, Shon, "Secure Communication using Embedding Drive Synchronization," J. of Korean Institute of Intelligent Systems, vol. 24, no. 6, Dec. 2014, pp. 615-621. https://doi.org/10.5391/JKIIS.2014.24.6.615
  5. S. Yu, C. Hyun, and M. Park, " Backstepping Control and Synchronization for 4-D Lorenz-Stenflo Chaotic System with Single Input," Int. J. of Fuzzy Logic and Intelligent Systems, vol. 11, no. 3, 2011, pp. 143-148. https://doi.org/10.5391/IJFIS.2011.11.3.143
  6. S. Yu, C. Hyun, and M. Park, "Control and Synchronization of New Hyperchaotic System using Active Backstepping Design," Int. J. of Fuzzy Logic and Intelligent Systems, vol. 11, no. 2, 2011, pp. 77-83. https://doi.org/10.5391/IJFIS.2011.11.2.077
  7. Y. Bae, "Diagnosis of power supply by analysis of chaotic nonlinear dynamics," J. of the Korea Institute of Electronic Communication Sciences, vol. 8, no. 1, 2013, pp. 113-119. https://doi.org/10.13067/JKIECS.2013.8.1.113
  8. Y. Bae, "Chaotic Phenomena in MEMS with Duffing Equation," J. of the Korea Institute of Electronic Communication Sciences, vol. 6, no. 6, 2011, pp. 709-716.
  9. Y. Bae and J. Park "A Study on Obstacle Avoid Method and Synchronization of multi chaotic robot for Robot Formation Control based on Chaotic Theory," J. of the Korea Institute of Electronic Communication Sciences, vol. 5, no. 5, 2010, pp. 534-540.
  10. Y. Bae, "A study on chaotic phenomenon in rolling mill bearing," J. of Korean Institute of Intelligent Systems, vol. 11, no. 4, Aug. 2001, pp. 315-319.
  11. Y. Bae, J. Kim, Y. Kim, and Y. Shon, "Secure communication using embedding drive synchronization," J. of Korean Institute of Intelligent Systems, vol. 13, no. 3, June 2003, pp. 310-315. https://doi.org/10.5391/JKIIS.2003.13.3.310
  12. Y. Bae, "Chaotic Phenomena in Addiction Model for Digital Leisure," Int. J. of Fuzzy Logic and Intelligent Systems vol. 13, no. 4, Dec. 2013, pp. 291-297. https://doi.org/10.5391/IJFIS.2013.13.4.291
  13. S. Metin and N. S. Sengor, "Dynamical system approach in modeling addiction," In Int. Conf. of Brain Inspired Cognitive Systems, Madrid, Spain, July 2010.
  14. M. Kim and Y. Bae, "Mathematical Modelling and Chaotic Behavior Analysis of Cyber Addiction," J. of Korean Institute of Intelligent Systems, vol. 24, no. 3, June 2014, pp. 245-250. https://doi.org/10.5391/JKIIS.2014.24.3.245
  15. Y. Bae, " Synchronization of Dynamical Happiness Model," Int. J. of Fuzzy Logic and Intelligent Systems, vol. 13, no. 4, 2013, pp. 291-297. https://doi.org/10.5391/IJFIS.2013.13.4.291
  16. S. Kim, S. Choi, Y. Bae, and Y. Park, "Mathematical Modelling of Happiness and its Nonlinear Analysis," J. of the Korea Institute of Electronic Communication Science, vol. 9, no. 6, 2014, pp. 711-717. https://doi.org/10.13067/JKIECS.2014.9.6.711
  17. Y. Bae, "Mathematical Modelling of Love and its Nonlinear," J. of the Korea Institute of Electronic Communication Science, vol. 9, no. 11, pp. 1297-1303, 2014. https://doi.org/10.13067/JKIECS.2014.9.11.1297
  18. Y. Bae, "Mathematical Modelling and Behavior Analysis of Addiction of Physical Exercise," J. of Korean Institute of Intelligent Systems, vol. 24, no. 6, Dec. 2014, pp. 615-621. https://doi.org/10.5391/JKIIS.2014.24.6.615
  19. Y. Bae, "Chaotic Dynamics in Tobacco's Addiction Model," Int. J. of Fuzzy Logic and Intelligent Systems, vol. 14, no. 4, 2014, pp. 322-331. https://doi.org/10.5391/IJFIS.2014.14.4.322
  20. F. C. Moon, Chaotic Vibrations. New York: Wiley, 1987.
  21. Y. H. Ku and X. Sun, "Chaos in Van der Pol's equation," J. of the Franklin Institute, vol. 327, no. 2, 1990, pp. 197-207. https://doi.org/10.1016/0016-0032(90)90016-C
  22. R. Magana. A, Hermosillo, and M. Perez "Fractional Numerical Methods in Geotechnics, 8th. World Congress on Computational Mechanics (WCCM8)," European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008), Venice, Italy, June 30 -July 5, 2008.
  23. K. Diethelm, The Analysis of Fractional Differential Equations. New York: Spriger, 2004.