Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

A Poincaré-Birkhoff theorem for tight Reeb flows on $S^3$

We consider Reeb flows on the tight $3$-sphere admitting a pair of closed orbits forming a Hopf link. If the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition then there exist infinitely many periodic trajectories distinguished by their linking numbers with the components of the link. This result admits a natural comparison to the Poincaré-Birkhoff theorem on area-preserving annulus homeomorphisms. An analogous theorem holds on $SO(3)$ and applies to geodesic flows of Finsler metrics on $S^2$.

Cotangent bundle reduction and Poincaré-Birkhoff normal forms

In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincar{é}-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincar{é}-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.

A generalized Poincaré-Birkhoff theorem

We prove a generalization of the classical Poincaré--Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem (as proved by the authors in arXiv:2011.10386).

Periodic Solutions of a Forced Relativistic Pendulum via Twist Dynamics

We prove the existence of at least two geometrically different periodic solution with winding number N for the forced relativistic pendulum. The instability of a solution is also proved. The proof is topological and based on the version of the Poincaré-Birkhoff theorem by Franks. Moreover, with some restriction on the parameters, we prove the existence of twist dynamics.

An Elementary Proof of Poincaré's Last Geometric Theorem

It is shown that the Poincaré-Birkhoff fixed point theorem may be proven by extending the geometric approach originally devised by Henri Poincaré himself, along with several results from elementary differential topology. Beginning with an example application of the theorem, we proceed by systematically constructing and classifying a certain set of invariant curves and their critical points. This classification is then used to prove the correctness of a procedure which guarantees the existence of at least two fixed points for any twist map of the annulus admitting a positive integral invariant.

Bounded homeomorphisms of the open annulus

We prove a generalization of the Poincaré-Birkhoff theorem for the open annulus showing that if a homeomorphism satisfies a certain twist condition and the nonwandering set is connected, then there is a fixed point. Our main focus is the study of bounded homeomorphisms of the open annulus. We prove a fixed point theorem for bounded homeomorphisms and study the special case of those homeomorphisms possessing at most one fixed point. Lastly we use the existence of rational rotation numbers to prove the existence of periodic orbits.

Periodic bouncing solutions of the Lazer-Solimini equation with weak repulsive singularity

We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be catalogued according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré-Birkhoff Theorem.

具有解析 Birkhoff 正规形式的台球桌一般是 Gevrey 发散的

Billiard tables with analytic Birkhoff normal form are generically Gevrey divergent

本文研究保面积映射的解析正规形式问题，通过放宽对称性条件扩展台球类，采用形式幂级数方法证明一般参数下台球桌级数是 Gevrey 发散而非解析收敛的。

Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems

In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0,i_\infty$ at the origin and at infinity, has at least $|i_\infty-i_0|$ periodic solutions, and an additional one if $i_0$ is even. Our argument combines the Poincaré--Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.

拟周期可逆映射的不变曲线及其应用

Invariant curves of quasi-periodic reversible mappings and its application

本文研究拟周期可逆映射的不变曲线存在性，通过正规形定理和修正同调方程保证可逆性，并应用于非线性共振振荡器，探讨Birkhoff常数的影响。

Planar Hamiltonian systems: index theory and applications to the existence of subharmonics

We consider a planar Hamiltonian system of the type $Jz' = \nabla_z H(t,z)$, where $H: \mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}$ is a function periodic in the time variable, such that $\nabla_z H(t,0) \equiv 0$ and $\nabla_z H(t,z)$ is asymptotically linear for $\vert z \vert \to +\infty$. After revisiting the index theory for linear planar Hamiltonian systems, by using the Poincaré-Birkhoff fixed point theorem we prove that the above nonlinear system has subharmonic solutions of any order $k$ large enough, whenever the rotation numbers (or, equivalently, the mean Conley-Zehnder indices) …