Homotopical Dynamics IV: Hopf invariants and hamiltonian flows

On the rigidity of integrable deformations of twist maps and Hamiltonian flows

In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e., homotopically non-trivial invariant curves of the cylinder on which the dynamics is conjugate to a rotation) for any rational rotation number in an open interval (without any further assumption on its size), then the deformation must be trivial. Analogue phenomena for integrable Tonelli Hamiltonian flows in higher dimensions are discussed.

Continuum Hamiltonian Hopf Bifurcation II

Building on the development of [MOR13], bifurcation of unstable modes that emerge from continuous spectra in a class of infinite-dimensional noncanonical Hamiltonian systems is investigated. Of main interest is a bifurcation termed the continuum Hamiltonian Hopf (CHH) bifurcation, which is an infinite-dimensional analog of the usual Hamiltonian Hopf (HH) bifurcation. Necessary notions pertaining to spectra, structural stability, signature of the continuous spectra, and normal forms are described. The theory developed is applicable to a wide class of 2+1 noncanonical Hamiltonian matter models, …

双重哈密顿 Hopf 分岔：规范化与正规型不可积性

Double Hamiltonian Hopf Bifurcation: normalization and normal form non-integrability

本文研究了四自由度光滑哈密顿系统的双重 Hopf 分岔，推导了无理频率比下的正规型，通过二次积分约化证明所得二自由度系统对几乎所有系数不可积，并数值分析了特定系统的同宿解与分岔。

A Hopf variables view on the libration points dynamics

The dynamics about the libration points of the Hill problem is investigated analytically. In particular, the use of Lissajous variables and perturbation theory allows to reduce the problem to a one degree of freedom Hamiltonian depending on two physical parameters. The invariant manifolds structure of the Hill problem is then disclosed, yet accurate computations are limited to energy values close to that of the libration points.

哈密顿平衡点附近的非共振 Hopf 链环

Non-resonant Hopf Links Near a Hamiltonian Equilibrium Point

本文研究二自由度哈密顿系统平衡点附近周期轨道的存在性，提出基于 Birkhoff-Gustavson 正规式和非共振条件确保能量面上存在无穷多周期轨道的显式判据，并应用于空间等腰三体问题、Hill 月球问题和 Hénon-Heiles 系统。

The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding…

Helicity-type integral invariants for Hamiltonian systems

In this note, we consider generalizations of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems with one-and-a-half degrees of freedom and symplectic diffeomorphisms of a two-disk to itself.

Hamilton Hopf 分岔的一般开折中的混沌现象，重点讨论超越 Gascheau-Routh 质量比的限制性平面圆型三体问题

Chaotic phenomena in generic unfoldings of the Hamilton Hopf bifurcation with emphasis on the restricted planar circular 3-body problem beyond the Gascheau-Routh mass ratio

本文证明 Hamilton Hopf 奇点的一般开折在亚临界参数处存在横截同宿轨道，从而产生任意大拓扑熵的混沌动力学，并将该结果条件性地应用于限制性平面圆型三体问题中 Lagrangian 点 L4 附近的 Hamiltonian，条件取决于 Stokes 常数的值。

具有任意涡量的 $f$ 平面毛细-重力流的小振幅孤立波

Small-Amplitude Solitary Waves for $f$-Plane Capillary-Gravity Flows with Arbitrary Vorticity

针对赤道 $f$ 平面上具有任意涡量的二维毛细-重力流，通过可逆哈密顿空间动力学系统约化，识别共振曲线并得到对称孤立波族。

共形辛系统的可积性

Integrability for conformally symplectic systems

本文针对耗散的共形辛哈密顿流提出两种可积性定义，证明无共轭点的Tonelli哈密顿流自动满足Hopf可积性，并给出渐近Maslov指标结果。

接触动力学中的平衡与分支

Equilibria and bifurcations in contact dynamics

本文研究接触动力学中平衡点与分支的结构，揭示了主系数在特征值中的作用及不同类型的平衡点引发的分支现象。

$C^\infty$ symplectic invariants of parabolic orbits and flaps in integrable Hamiltonian systems

In the present paper, we consider a smooth $C^\infty$ symplectic classification of Lagrangian fibrations near cusp singularities, parabolic orbits and cuspidal tori. We show that for these singularities as well as for an arrangement of singularities known as a flap, which arises in the integrable subcritical Hamiltonian Hopf bifurcation, the action variables form a complete set of $C^\infty$ symplectic invariants. We also give a symplectic classification for parabolic orbits in the real-analytic case. Namely, we prove that a complete symplectic invariant in this case is given by a real-analyti…

三涡旋系统：Hopf纤维化、辛约化与近碰撞

The three-vortex system: Hopf fibration, symplectic reduction, and near-collisions

本文研究平面三涡旋系统的碰撞与近碰撞动力学，通过约化坐标和辛约化方法分析，提出新的近碰撞结果并证明新的临界能量不等式，同时重现经典结论。