| Title:
|
Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation (English) |
| Author:
|
Mizeal, Ahmed Abbas |
| Author:
|
Hussain, Mudhir A. Abdul |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
27-37 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
bifurcation theory |
| Keyword:
|
nonlinear systems |
| Keyword:
|
local Lyapunov-Schmidt method |
| MSC:
|
34K18 |
| MSC:
|
93C10 |
| idMR:
|
MR2915847 |
| DOI:
|
10.5817/AM2012-1-27 |
| . |
| Date available:
|
2012-03-15T18:07:21Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142089 |
| . |
| Reference:
|
[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. |
| Reference:
|
[2] Abdul Hussain, M. A.: Bifurcation solutions of elastic beams equation with small perturbation.Int. J. Math. Anal. (Ruse) 3 (18) (2009), 879–888. Zbl 1195.74093, MR 2604465 |
| Reference:
|
[3] Arnol’d, V. I.: Singularities of differential maps.Math. Sci. (1989). |
| Reference:
|
[4] Ishibashi, Y. J.: Phenomenological theory of domain walls.Ferroelectrics 98 (1989), 193–205. 10.1080/00150198908217582 |
| Reference:
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[5] Loginov, B. V.: Theory of Branching nonlinear equations in the conditions of invariance group.Fan, Tashkent (1985). MR 0878356 |
| Reference:
|
[6] Sapronov, Y. I.: Regular perturbation of Fredholm maps and theorem about odd field.Works Dept. of Math., Voronezh Univ. 10 (1973), 82–88. |
| Reference:
|
[7] Sapronov, Y. I.: Nonlocal finite dimensional reduction in the variational boundary value problems.Mat. Zametki 49 (1991), 94–103. MR 1101555 |
| Reference:
|
[8] Sapronov, Y. I., Darinskii, B. M., Tcarev, C. L.: Bifurcation of extremely of Fredholm functionals.Voronezh Univ. (2004). |
| Reference:
|
[9] Sapronov, Y. I., Zachepa, V. R.: Local analysis of Fredholm equation.Voronezh Univ. (2002). |
| Reference:
|
[10] Thompson, J. M. T., Stewart, H. B.: Nonlinear Dynamics and Chaos.Chichester, Singapore, J. Wiley and Sons, 1986. Zbl 0601.58001, MR 0854476 |
| Reference:
|
[11] Vainbergm, M. M., Trenogin, V. A.: Theory of branching solutions of nonlinear equations.Math. Sci. (1969). MR 0261416 |
| . |
