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Generalized Lyapunov-Schmidt reduction method and normal forms for the study of bifurcations of periodic points in families of reversible diffeomorphisms

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Abstract
We introduce a general reduction method for the study of periodic points near a fixed point in a family of reversible diffeomorphisms. We impose no restrictions on the linearization at the fixed point except invertibility, allowing higher multiplicities. It is shown that the problem reduces to a similar problem for a reduced family of diffeomorphisms, which is itself reversible, but also has an additional Z(q)-symmetry. The reversibility in combination with the Z(q)-symmetry translates to a D-q-symmetry for the problem, which allows to write down the bifurcation equations. Moreover, the reduced family can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Lyapunov-Schmidt method, and makes repetitive use of the Implicit Function Theorem. As an application we analyze the branching of periodic points near a fixed point in a family of reversible mappings, when for a critical value of the parameters the linearization at the fixed point has either a pair of simple purely imaginary eigenvalues that are roots of unity or a pair of non-semisimple purely imaginary eigenvalues that are roots of unity with algebraic multiplicity 2 and geometric multiplicity 1.
Keywords
generalized Lyapunov-Schmidt reduction, DYNAMICAL-SYSTEMS, normal forms, reversible maps, periodic orbits, SYMMETRY, FLOWS, MAPS

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Chicago
Ciocci, Maria-Cristina. 2004. “Generalized Lyapunov-Schmidt Reduction Method and Normal Forms for the Study of Bifurcations of Periodic Points in Families of Reversible Diffeomorphisms.” Journal of Difference Equations and Applications 10 (7): 621–649.
APA
Ciocci, M.-C. (2004). Generalized Lyapunov-Schmidt reduction method and normal forms for the study of bifurcations of periodic points in families of reversible diffeomorphisms. JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, 10(7), 621–649.
Vancouver
1.
Ciocci M-C. Generalized Lyapunov-Schmidt reduction method and normal forms for the study of bifurcations of periodic points in families of reversible diffeomorphisms. JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS. 2004;10(7):621–49.
MLA
Ciocci, Maria-Cristina. “Generalized Lyapunov-Schmidt Reduction Method and Normal Forms for the Study of Bifurcations of Periodic Points in Families of Reversible Diffeomorphisms.” JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 10.7 (2004): 621–649. Print.
@article{1200781,
  abstract     = {We introduce a general reduction method for the study of periodic points near a fixed point in a family of reversible diffeomorphisms. We impose no restrictions on the linearization at the fixed point except invertibility, allowing higher multiplicities. It is shown that the problem reduces to a similar problem for a reduced family of diffeomorphisms, which is itself reversible, but also has an additional Z(q)-symmetry. The reversibility in combination with the Z(q)-symmetry translates to a D-q-symmetry for the problem, which allows to write down the bifurcation equations. Moreover, the reduced family can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Lyapunov-Schmidt method, and makes repetitive use of the Implicit Function Theorem. As an application we analyze the branching of periodic points near a fixed point in a family of reversible mappings, when for a critical value of the parameters the linearization at the fixed point has either a pair of simple purely imaginary eigenvalues that are roots of unity or a pair of non-semisimple purely imaginary eigenvalues that are roots of unity with algebraic multiplicity 2 and geometric multiplicity 1.},
  author       = {Ciocci, Maria-Cristina},
  issn         = {1023-6198},
  journal      = {JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS},
  keyword      = {generalized Lyapunov-Schmidt reduction,DYNAMICAL-SYSTEMS,normal forms,reversible maps,periodic orbits,SYMMETRY,FLOWS,MAPS},
  language     = {eng},
  number       = {7},
  pages        = {621--649},
  title        = {Generalized Lyapunov-Schmidt reduction method and normal forms for the study of bifurcations of periodic points in families of reversible diffeomorphisms},
  url          = {http://dx.doi.org/10.1080/10236190410001647816},
  volume       = {10},
  year         = {2004},
}

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