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

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

DOI QR Code

Analysis of Nonlinear Behavior in Fractional Van der Pol Equation with Periodic External Force and Fractional Differential Equation

분수 차수 미분 방정식과 주기적인 외력을 가진 Van der Pol 발진기에서의 비선형 거동 해석

  • Received : 2015.12.23
  • Accepted : 2016.02.24
  • Published : 2016.02.25
Download PDF
⟨ Previous Next ⟩

Abstract

Van der Pol's oscillators is non-conservative oscillator that having nonlinear damping phenomena. The energy of its system is dissipative at a high amplitude whereas its system creates the energy at low amplitude. This paper deals with the Van der Pol oscillator model with a fractional order when the external force apply into Van der Pol oscillator. This paper confirms the status of variation for the limit cycle according to the parameter variation of fractional order in the Van der Pol oscillator that can be represented by fractional differential equation.

Van der Pol 발진기는 비선형 제동 현상을 가진 비보존 발진기로서 높은 진폭에서의 에너지는 소산적이며 낮은 진폭들에서는 생성되는 구조를 가진다. 본 논문에서는 분수 차수를 가지는 Van der Pol 발진기 모델에서 주기적 외력을 인가하였을 경우 분수차수로 표현되는 미분 방정식에서 분수차수의 파라미터 변화에 따른 리미트 사이클이 변화 상태를 확인하고자 한다.

Keywords

References

  1. J. J. Slotine and W. Li, Applied Nonlinear Control. 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. 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," 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. C. Bae, J. W. Kim, Y. G. Kim, and Y. W. 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. 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: Springer, 2004.