The Analysis of Bifurcation Solutions by Angular Singularities

Authors

  • Hussein K. Kadhim Department of Mathematics, Faculty of Education for Pure Sciences, University of Basrah, Basrah
  • Mudhir A. Abdul Hussain Department of Mathematics, Faculty of Education for Pure Sciences, University of Basrah, Basrah

DOI:

https://doi.org/10.26713/cma.v10i4.1250

Keywords:

Bifurcation solutions, Angular Singularities, Caustic

Abstract

This paper studies a nonlinear wave equation's bifurcation solutions of elastic beams situated on elastic bases with small perturbation by using the local method of Lyapunov-Schmidt. We have found the Key function corresponding to the functional related to this equation. The bifurcation analysis of this function has been investigated by the angular singularities. We have found the parametric equation of the bifurcation set (caustic) with the geometric description of this caustic. Also, the critical points' bifurcation spreading has been found.

Downloads

Download data is not yet available.

References

M. A. Abdul Hussain, 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) (in Russian).

V. I. Arnold, Singularities of Differential Maps, M. Science (1989).

G. Berczi, Lectures on Singularities of Maps, Trinity Term, Oxford (2010).

B. M. Darinskii, C. L. Tcarev and Yu. I. Sapronov, Bifurcations of extremals of Fredholm functionals, Journal of Mathematical Sciences 145(6) (2007).

E. V. Derunova and Yu. I. Sapronov, Application of normalized key functions in a problem of branching of periodic extremals, Russian Mathematics (Iz. Vuz) 59(8) (2015).

M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory: Vol. I (Applied Mathematical Sciences 51), Springer, New York (1985), DOI: 10.1007/978-1-4612-5034-0.

B. V. Loginov, Theory of Branching Nonlinear Equations in The Conditions of Invariance Group, Fan, Tashkent (1985).

Yu. I. Sapronov, Regular perturbation of Fredholm maps and theorem about odd field, Works Dept. of Math., Voronezh University, 10, 82 – 88 (1973).

Yu. I. Sapronov, Finite dimensional reduction in the smooth extremely problems, Uspehi Math. Science 51(1) (1996), 101 – 132.

Yu. I. Sapronov and V. R. Zachepa, Local Analysis of Fredholm Equation, Voronezh University (2002) (in Russian).

Yu. I. Sapronov, B. M. Darinskii and C. L. Tcarev, Bifurcation of Extremely of Fredholm Functionals, Voronezh University (2004).

O. V. Shveriova, Caustic and bif-spreadings for a boundary extremal with a triple degeneration along the boundary, Tr. Mat. Fak. Voronezh. Gos. Univ. (N.S.) 7 (2002), 149 – 160 (in Russian).

M. M. Vainberg and V. A. Trenogin, Theory of Branching Solutions of Non-linear Equations, M. Science (1969), DOI: 10.1137/1018033.

Downloads

Published

31-12-2019
CITATION

How to Cite

Kadhim, H. K., & Hussain, M. A. A. (2019). The Analysis of Bifurcation Solutions by Angular Singularities. Communications in Mathematics and Applications, 10(4), 733–744. https://doi.org/10.26713/cma.v10i4.1250

Issue

Vol. 10 No. 4 (2019)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.