On the Structure of the Set of Bifurcation Points of Periodic Solutions for Multiparameter Hamiltonian Systems

This paper deals with periodic solutions of the Hamilton equation ẋ(t) = J∇xH(x(t), λ), where H ∈ C2,0(R2n × Rk,R) and λ ∈ Rk is a parameter. Theorems on global bifurcation of solutions with periods 2π j , j ∈ N, from a stationary point (x0, λ0) ∈ R2n × Rk are proved. ∇xH(x0, λ0) can be singular. However, it is assumed that the local topological degree of ∇xH(·, λ0) at x0 is nonzero. For systems satisfying ∇xH(x0, λ) = 0 for all λ ∈ Rk it is shown that (global) bifurcation points of solutions with periods 2π j can be identified with zeros of appropriate continuous functions Fj : Rk → R. If, for all λ ∈ Rk, ∇xH(x0, λ) = diag(A(λ), B(λ)), where A(λ) and B(λ) are (n×n)-matrices, then Fj can be defined by Fj(λ) = det[A(λ)B(λ)− j2I]. 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, Toruń, 2005].