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On the Butterfly Catastrophe Model and Stability of Finite Periodic Solutions for Some Non-Linear Differential Equations

    Author

    • Isam R. Faeq

    Computer Engineering Department, Technical College of Kirkuk, Northern, Kirkuk,iraq

,

Document Type : Research Paper

10.32894/kujss.2023.136973.1089
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Abstract

 In this work, we find the results for the folded part projection of the butterfly catastrophe model onto the control space, using methods from catastrophe theory to obtain stability and the catastrophic behavior of finite periodic solutions for some non-linear differential equations. Finally, we have shown that a saddle-node bifurcation, which can be classified as a butterfly mutation, accompanies butterfly surface folding.

Keywords

  • Butterfly catastrophe model
  • butterfly type catastrophe
  • non-linear differential equations
  • limit cycles

Main Subjects

  • Differential equation
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References
[1] M.N. Murad Kaki. On the cusp catastrophe model and stability. General Mathematics Notes, 2(2): 73–82, 2011.
[2] K. D. Arrow Smith and K. L. Taha. Vector Fields Mecanica. 18, 1983.
[3] L. Cesari. Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Academic Press, New York, 2nd revised edition, 1963, doi:10.1090/S0002-99391960-0121542-7.
[4] P. Hartman. A lemma in the theory of structural stability of differential equations. Proceedings of the American Mathematical Society, 14(1963): 568–578, 1963.
[5] K. L. Hale. Ordinary Differential Equations. John Wiley and Sons, New York, 2nd edition, 1960.
[6] W. Hirsch and S. Smile. Differential equations, dynamical systems and linear algebra. 1974.
[7] C. Hayashi. Nonlinear Oscillations in Physical Systems. McGraw Hill, New York, (1964), Reissue, Princeton Univ. Press (1984).
[8] E. J. Marsden and M. McCracken. The Hopf Bifurcation and its Applications, volume 19. Springer-Verlag, New York, Heidelberg Berlin, 1976.
[9] M.N. Mohammad. Treatment of Phenomena of Instability by Method of Catastrophe Theory. Master’s thesis, University of Baghdad, Baghdad, Iraq, 1985.
[10] M. N. Murad Kaki. Mathematical catastrophe with applications. General Mathematics Notes, 11(2): 35–46, 2012.
[11] W. D. Jordon and P. Smith. Nonlinear Ordinary Differential Equation. Oxford University Press, 2nd  edition, 1989.
[12] E. C. Zeeman. Catastrophe theory: Selected papers. 1977.
[13] M. N. Murad Kaki. Non-linear dynamics and cusp catastrophe. Journal of College of Education, University of Al-Mustansiriya, (5): 101–110, 2013.
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Kirkuk University Journal-Scientific Studies
Volume 18, Issue 1
March 2023
Page 31-34
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  • Article View: 72
  • PDF Download: 62

APA

Faeq, I. R. (2023). On the Butterfly Catastrophe Model and Stability of Finite Periodic Solutions for Some Non-Linear Differential Equations. Kirkuk University Journal-Scientific Studies, 18(1), 31-34. doi: 10.32894/kujss.2023.136973.1089

MLA

Isam R. Faeq. "On the Butterfly Catastrophe Model and Stability of Finite Periodic Solutions for Some Non-Linear Differential Equations". Kirkuk University Journal-Scientific Studies, 18, 1, 2023, 31-34. doi: 10.32894/kujss.2023.136973.1089

HARVARD

Faeq, I. R. (2023). 'On the Butterfly Catastrophe Model and Stability of Finite Periodic Solutions for Some Non-Linear Differential Equations', Kirkuk University Journal-Scientific Studies, 18(1), pp. 31-34. doi: 10.32894/kujss.2023.136973.1089

VANCOUVER

Faeq, I. R. On the Butterfly Catastrophe Model and Stability of Finite Periodic Solutions for Some Non-Linear Differential Equations. Kirkuk University Journal-Scientific Studies, 2023; 18(1): 31-34. doi: 10.32894/kujss.2023.136973.1089

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