한국전자통신학회논문지 (The Journal of the Korea institute of electronic communication sciences)

한국전자통신학회 (Korea Institute of Electronic Communication Science)
권호 표지
DOI QR코드

DOI QR Code

Fractional Duffing 방정식에서의 카오스 거동 해석

Analysis of Chaotic Behavior in Fractional Duffing Equation

  • 배영철 (전남대학교 전기.전자통신.컴퓨터공학부)
  • Bae, Young-Chul (Division of Electrical.Electronics Communication and Computer Engineering, Chonnam National University)
  • 투고 : 2015.11.10
  • 심사 : 2015.12.24
  • 발행 : 2015.12.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

최근에 fractional calculus의 개념을 적용하여 fractional 미분 방정식으로 표현되는 기법이 제어공학, 수학, 물리학 등에 적용하고자 하는 노력이 나타나고 있다. 본 논문에서는 Duffing 방정식으로 표현되는 동적 방정식을 정수 차수가 아닌 fractional 차수로 표현하고 이 fractional 실수 차수에서 차수의 크기에 따라 카오스 거동이 존재함을 실수 차수의 값을 변화시켜가면서 시계열 데이터와 위상공간으로 확인하고자 한다.

Recently many effort appears applying the concept of fractional calculus that can be represented by fractional differential equation in the control engineering, physics and mathematics. This paper describes the fractional order with real order for Duffing equation which can be represented by integer order. This paper also confirms the existence of chaotic behaviors by using time series and phase portrait with varying the parameter of real order.

키워드

참고문헌

  1. J. J. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, NJ, 1991.
  2. J. K. Hale, Oscillations in Nonlinear Systems, McGraw-Hill, New York, 1963.
  3. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 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, December 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," International Journal of Fuzzy Logic and Intelligent Systems, vol. 11, no. 3, 2011, pp. 135-216. https://doi.org/10.5391/IJFIS.2011.11.3.135
  6. S. Yu, C. Hyun and M. Park, "Control and Synchronization of New Hyperchaotic System using Active Backstepping Design," International Journal 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, 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, pp. 534-540, 2010.
  10. Y. C. Bae, " A study on chaotic phenomenon in rolling mill bearing," Journal of Korean Institute of Intelligent Systems, vol. 11, no. 4, Aug. 2001, pp. 315-319.
  11. Y. C. Bae, J. W. Kim, Y. G. Kim, and Y. W. Shon, "Secure communication using embedding drive synchronization," Journal of Korean Institute of Intelligent Systems, vol. 13, no. 3, June 2003, pp. 310-315. http://dx.doi.org/10.5391/JKIIS.2003.13.3.310
  12. Y. Bae, "Chaotic Phenomena in Addiction Model for Digital Leisure," International Journal of Fuzzy Logic and Intelligent Systems vol. 13, no. 4, December 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 International Conference of Brain Inspired Cognitive Systems, Madrid, Spain, July 14-16, 2010.
  14. M. Kim, Y. Bae, "Mathematical Modelling and Chaotic Behavior Analysis of Cyber Addiction,"Journal of Korean Institute of Intelligent Systems, vol. 24, no. 3, Jun. 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," Journal 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, Wiley, New York, 1987.
  21. P. J. Holmes, D. A. Rand, "The bifurcations of Duffing's equation: an application of catastrophe theory," Journal of Sound and Vibration, 44, 1976, pp. 237-253. https://doi.org/10.1016/0022-460X(76)90771-9
  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), June 30 - July 5, 2008, Venice, Italy.
  23. W. Eric, "Fractional Calculus", From MathWorld-A Wolfram Web Resource.