About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Böhmer, Klaus ; Janovská, Drahoslava ; Janovský, Vladimír
Computing the differential of an unfolded contact diffeomorphism. (English). Applications of Mathematics, vol. 48 (2003), issue 1, pp. 3-30
MSC: 34A34, 35B32, 37C05, 47J15, 58K20, 65L99, 65P30 | MR 1954501 | Zbl 1099.34007 | DOI: 10.1023/A:1022950819918
Keywords:
bifurcation points; imperfect bifurcation diagrams; qualitative analysis
Summary:
Consider a bifurcation problem, namely, its bifurcation equation. There is a diffeomorphism $\Phi $ linking the actual solution set with an unfolded normal form of the bifurcation equation. The differential $D\Phi (0)$ of this diffeomorphism is a valuable information for a numerical analysis of the imperfect bifurcation. The aim of this paper is to construct algorithms for a computation of $D\Phi (0)$. Singularity classes containing bifurcation points with $\mathop {\mathrm codim}\le 3$, $\mathop {\mathrm corank}=1$ are considered.
References:
[1] K. Böhmer: On a numerical Lyapunov-Schmidt method for operator equations. Computing 53 (1993), 237–269. MR 1253405
[2] K.  Böhmer, D.  Janovská and V.  Janovský: Computer aided analysis of the imperfect bifurcation diagrams. East-West J. Numer. Math. (1998), 207–222. MR 1652813
[3] K. Böhmer, D. Janovská and V. Janovský: On the numerical analysis of the imperfect bifurcation. SIAM J. Numer. Anal. 40 (2002), 416–430. DOI 10.1137/S0036142900369283 | MR 1921663
[4] S. N. Chow, J.  Hale: Methods of Bifurcation Theory. Springer Verlag, New York, 1982. MR 0660633
[5] M. Golubitsky, D.  Schaeffer: A theory for imperfect bifurcation via singularity theory. Commun. Pure Appl. Math. 32 (1979), 21–98. DOI 10.1002/cpa.3160320103 | MR 0508917
[6] M. Golubitsky, D.  Schaeffer: Singularities and Groups in Bifurcation Theory, Vol. 1. Springer Verlag, New York, 1985. MR 0771477
[7] W. Govaerts: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia, 2000. MR 1736704 | Zbl 0935.37054
[8] V. Janovský, P.  Plecháč: Computer aided analysis of imperfect bifurcation diagrams I. Simple bifurcation point and isola formation centre. SIAM J.  Num. Anal. 21 (1992), 498-512. MR 1154278
Partner of
EuDML logo