Computing the differential of an unfolded contact diffeomorphism

Klaus Böhmer; Drahoslava Janovská; Vladimír Janovský

Applications of Mathematics (2003)

  • Volume: 48, Issue: 1, page 3-30
  • ISSN: 0862-7940

Abstract

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Consider a bifurcation problem, namely, its bifurcation equation. There is a diffeomorphism Φ linking the actual solution set with an unfolded normal form of the bifurcation equation. The differential D Φ ( 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 Φ ( 0 ) . Singularity classes containing bifurcation points with c o d i m 3 , c o r a n k = 1 are considered.

How to cite

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Böhmer, Klaus, Janovská, Drahoslava, and Janovský, Vladimír. "Computing the differential of an unfolded contact diffeomorphism." Applications of Mathematics 48.1 (2003): 3-30. <http://eudml.org/doc/33131>.

@article{Böhmer2003,
abstract = {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 $\mathop \{\mathrm \{c\}odim\}\le 3$, $\mathop \{\mathrm \{c\}orank\}=1$ are considered.},
author = {Böhmer, Klaus, Janovská, Drahoslava, Janovský, Vladimír},
journal = {Applications of Mathematics},
keywords = {bifurcation points; imperfect bifurcation diagrams; qualitative analysis; bifurcation points; imperfect bifurcation diagrams; qualitative analysis},
language = {eng},
number = {1},
pages = {3-30},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Computing the differential of an unfolded contact diffeomorphism},
url = {http://eudml.org/doc/33131},
volume = {48},
year = {2003},
}

TY - JOUR
AU - Böhmer, Klaus
AU - Janovská, Drahoslava
AU - Janovský, Vladimír
TI - Computing the differential of an unfolded contact diffeomorphism
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 1
SP - 3
EP - 30
AB - 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 $\mathop {\mathrm {c}odim}\le 3$, $\mathop {\mathrm {c}orank}=1$ are considered.
LA - eng
KW - bifurcation points; imperfect bifurcation diagrams; qualitative analysis; bifurcation points; imperfect bifurcation diagrams; qualitative analysis
UR - http://eudml.org/doc/33131
ER -

References

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  2. Computer aided analysis of the imperfect bifurcation diagrams, East-West J. Numer. Math. (1998), 207–222. (1998) MR1652813
  3. 10.1137/S0036142900369283, SIAM J. Numer. Anal. 40 (2002), 416–430. (2002) MR1921663DOI10.1137/S0036142900369283
  4. Methods of Bifurcation Theory, Springer Verlag, New York, 1982. (1982) MR0660633
  5. 10.1002/cpa.3160320103, Commun. Pure Appl. Math. 32 (1979), 21–98. (1979) MR0508917DOI10.1002/cpa.3160320103
  6. Singularities and Groups in Bifurcation Theory, Vol. 1, Springer Verlag, New York, 1985. (1985) MR0771477
  7. Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000. (2000) Zbl0935.37054MR1736704
  8. Computer aided analysis of imperfect bifurcation diagrams I. Simple bifurcation point and isola formation centre, SIAM J.  Num. Anal. 21 (1992), 498-512. (1992) MR1154278

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