On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

Details On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

2008-09-28

arxiv.org/abs/0809.4858v1

Nonlinearity 23 (8) (2010) 1771-1806

Mathematics

Classical Analysis and ODEs

35 pages, 4 figures, PDFLaTeX

Scientific paper

10.1088/0951-7715/23/8/002

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].

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