@phdthesis{oai:sucra.repo.nii.ac.jp:00019055,
 author = {ATIA, AFROZ},
 month = {},
 note = {xiii, 71 p., We consider buckling of rod which is subjected to compressive force λ. In 1757, L. Euler’s found the critical load of the system, and this problem is often called Euler buckling problem. This is actually a celebrated example of pitchfork bifurcation. M. Golubitsky and D. Schaeffer considered a modified version of Euler buckling problem in the variational formulation using strain energy and potential energy, and they show that this modified problem is a versal unfolding of the original problem. We consider a rather more general problem in the context of variational set-up and discuss smoothness of the problem, which is not discussed by M. Golubitsky and D. Schaeffer. This is important to apply Lyapunov-Schmidt reduction. We also describe 3-jets of the equations which define the bifurcation set B and the hysteresis set H, which enable us to draw figures of B and H approximately under suitable set-up., Declaration of Authorship iii
Abstract v
Acknowledgements vii
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminaries 5
2.1 Sobolev Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Versal Unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 P-K equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Basic Bifurcation theorem . . . . . . . . . . . . . . . . . . . . . 9
2.5 Bifurcation Set and Hysteresis Set . . . . . . . . . . . . . . . . . 10
3 Euler Buckling Problem 13
3.1 Buckling of the rod with pinned ends . . . . . . . . . . . . . . 13
3.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Energy formulation of Euler buckling problem . . . . . . . . . 15
3.4 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Smoothness of Φ 19
4.1 Differentiability of Φ . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Φ is C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Φ is C∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Taylor coefficients of Φ 29
5.1 Taylor expansion of Φ . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 First order derivative of (L)u . . . . . . . . . . . . . . . . . . . 30
5.3 Derivatives of the coefficient of α1 . . . . . . . . . . . . . . . . 31
5.4 Second order derivative of (L)u . . . . . . . . . . . . . . . . . . 32
5.5 Third order derivative of (L)u . . . . . . . . . . . . . . . . . . . 32
6 Lyapunov-Schmidt reduction 35
6.1 The first order derivatives of W . . . . . . . . . . . . . . . . . . 36
6.2 The second order derivatives of W . . . . . . . . . . . . . . . . 37
7 Bifurcation equation F = 0 and its Taylor coefficients 41
7.1 The first order derivatives of F . . . . . . . . . . . . . . . . . . 41
7.2 The second order derivatives of F . . . . . . . . . . . . . . . . . 42
7.3 The third order derivatives of F . . . . . . . . . . . . . . . . . . 42
8 Versality 45
8.1 P-K-versality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9 Bifurcation set and hysteresis set 47
9.1 Equation of B and H . . . . . . . . . . . . . . . . . . . . . . . . 47
9.2 Figures of B and H . . . . . . . . . . . . . . . . . . . . . . . . . 52
A Appendix 63
A.1 Inverse Function Theorem and Implicit Function Theorem . . 63
A.2 Second variation of Energy equation of Euler buckling problem 63
A.3 Reduction Method of Lyapunov-Schmidt . . . . . . . . . . . . 68
Bibliography 71, 主指導教員 : 福井敏純, text, application/pdf},
 school = {埼玉大学},
 title = {Bifurcation Analysis of Euler Buckling Problem from the viewpoint of Singularity Theory},
 year = {2019},
 yomi = {アティア, アフロズ}
}