Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0809.4858

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

Wiktor Radzki
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2π/j,$ $j\in\mathbb{N},$ 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, Toruń, 2005].

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wiktor Radzki. 2008-09-28. On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems. https://doi.org/10.1088/0951-7715%2F23%2F8%2F002

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq \, 34/11$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗
arXiv2026

Dimension-free estimates for discrete maximal functions over cubes in $\mathbb Z^d$

In this short note, we establish dimension-free $\ell^p(\mathbb Z^d)$ bounds, for all $p\in(1,\infty]$, for the discrete Hardy--Littlewood maximal functions associated with cubes in $\mathbb Z^d$, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the $\ell^p(\mathbb Z^d)$ norms of the differences of the corresponding averages. This follows from an ad hoc interpretation of the associated discrete multipliers as a special continuous family of multipliers to which basic fractional integration and complex interpolation can be applied. The same method also yields an elementary proof of Bourgain's dimension-free $L^p(\mathbb R^d)$ bounds for the Hardy--Littlewood maximal function associated with cubes in $\mathbb R^d$.

math.CA↗
arXiv2026

Establishing the Polynomial Wolff Axioms for $δ$-Separated $δ$-Tubes With #o-minimality

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $δ$-separated $δ$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

math.CA↗