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

TL;DR

This paper investigates the structure of bifurcation points of periodic solutions in multiparameter Hamiltonian systems, providing theoretical results, explicit formulas, symmetry breaking insights, and practical applications for analyzing these bifurcations.

Contribution

It introduces new theorems on global bifurcation points, explicit formulas for bifurcation functions, and symmetry breaking results, advancing understanding of Hamiltonian system bifurcations.

Findings

Identification of bifurcation points with zeros of continuous functions

Explicit formulas for bifurcation functions in block-diagonal Hessians

Symmetry breaking results for solutions with different periods

Abstract

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].