The fold-flip bifurcation

Yu A. Kuznetsov; H. G.E. Meijer; L. Van Veen

doi:10.1142/S0218127404010576

The fold-flip bifurcation

Yu A. Kuznetsov^*
, H. G.E. Meijer
, L. Van Veen

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

80 Citations (Scopus)

Abstract

The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Hénon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincaré map are computed using variational equations to find the normal form coefficients.

Original language	English
Pages (from-to)	2253-2282
Number of pages	30
Journal	International journal of bifurcation and chaos in applied sciences and engineering
Volume	14
Issue number	7
DOIs	https://doi.org/10.1142/S0218127404010576
Publication status	Published - 1 Jan 2004
Externally published	Yes

Keywords

Bifurcations of fixed points
Center manifold
Normal forms

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AU - Meijer, H. G.E.

AU - Van Veen, L.

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N2 - The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Hénon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincaré map are computed using variational equations to find the normal form coefficients.

AB - The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Hénon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincaré map are computed using variational equations to find the normal form coefficients.

KW - Bifurcations of fixed points

KW - Center manifold

KW - Normal forms

UR - https://www.scopus.com/pages/publications/13844255362

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M3 - Article

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JO - International journal of bifurcation and chaos in applied sciences and engineering

JF - International journal of bifurcation and chaos in applied sciences and engineering

IS - 7

ER -

The fold-flip bifurcation

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Keywords

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