Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation

Ahmed Abbas Mizeal; Mudhir A. Abdul Hussain

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 1, page 27-37
  • ISSN: 0044-8753

Abstract

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In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.

How to cite

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Mizeal, Ahmed Abbas, and Hussain, Mudhir A. Abdul. "Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation." Archivum Mathematicum 048.1 (2012): 27-37. <http://eudml.org/doc/246547>.

@article{Mizeal2012,
abstract = {In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.},
author = {Mizeal, Ahmed Abbas, Hussain, Mudhir A. Abdul},
journal = {Archivum Mathematicum},
keywords = {bifurcation theory; nonlinear systems; local Lyapunov-Schmidt method; bifurcation theory; nonlinear system; local Lyapunov-Schmidt method},
language = {eng},
number = {1},
pages = {27-37},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation},
url = {http://eudml.org/doc/246547},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Mizeal, Ahmed Abbas
AU - Hussain, Mudhir A. Abdul
TI - Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 1
SP - 27
EP - 37
AB - In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.
LA - eng
KW - bifurcation theory; nonlinear systems; local Lyapunov-Schmidt method; bifurcation theory; nonlinear system; local Lyapunov-Schmidt method
UR - http://eudml.org/doc/246547
ER -

References

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  1. Abdul Hussain, M. A., Corner singularities of smooth functions in the analysis of bifurcations balance of the elastic beams and periodic waves, Ph.D. thesis, Voronezh University, Russia., 2005. (2005) 
  2. Abdul Hussain, M. A., Bifurcation solutions of elastic beams equation with small perturbation, Int. J. Math. Anal. (Ruse) 3 (18) (2009), 879–888. (2009) Zbl1195.74093MR2604465
  3. Arnol’d, V. I., Singularities of differential maps, Math. Sci. (1989). (1989) 
  4. Ishibashi, Y. J., 10.1080/00150198908217582, Ferroelectrics 98 (1989), 193–205. (1989) DOI10.1080/00150198908217582
  5. Loginov, B. V., Theory of Branching nonlinear equations in the conditions of invariance group, Fan, Tashkent (1985). (1985) MR0878356
  6. Sapronov, Y. I., Regular perturbation of Fredholm maps and theorem about odd field, Works Dept. of Math., Voronezh Univ. 10 (1973), 82–88. (1973) 
  7. Sapronov, Y. I., Nonlocal finite dimensional reduction in the variational boundary value problems, Mat. Zametki 49 (1991), 94–103. (1991) MR1101555
  8. Sapronov, Y. I., Darinskii, B. M., Tcarev, C. L., Bifurcation of extremely of Fredholm functionals, Voronezh Univ. (2004). (2004) 
  9. Sapronov, Y. I., Zachepa, V. R., Local analysis of Fredholm equation, Voronezh Univ. (2002). (2002) 
  10. Thompson, J. M. T., Stewart, H. B., Nonlinear Dynamics and Chaos, Chichester, Singapore, J. Wiley and Sons, 1986. (1986) Zbl0601.58001MR0854476
  11. Vainbergm, M. M., Trenogin, V. A., Theory of branching solutions of nonlinear equations, Math. Sci. (1969). (1969) MR0261416

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