Relative periodic orbits in point vortex systems

准周期通道中的点涡动力学：输运轨迹与平衡态的遍历分布

Point vortex dynamics in quasi-periodic channels: transporting trajectories and ergodic distribution of equilibria

研究准周期通道中点涡的动力学，通过壳层公式构造输运轨迹，并利用遍历性建立临界点分布的零点计数定理，揭示其渐近密度与相位分布。

Vortex dynamics on a cylinder

Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a non-existence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging.

Stability of periodic solutions of the N-vortex problem in general domains

We investigate stability properties of a type of periodic solutions of the $N$-vortex problem on general domains $Ω\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point $a_0\inΩ$ of the Robin function associated to the Dirichlet Laplacian of $Ω$. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point $a…

Periodic solutions with prescribed minimal period of vortex type problem in domains

We consider Hamiltonian systems with two degrees of freedom of point vortex type \[ κ_j \dot{z}_j = J \nabla_{z_j} H_Ω(z_1,z_2), \quad j=1,2, \] for $z_1,z_2$ in a domain $Ω\subset\mathbb{R}^2$. In the classical point vortex context the Hamiltonian $H_Ω$ is of the form \[ H_Ω(z_1,z_2) = -\frac{κ_1 κ_2}π \log |z_1-z_2| - 2κ_1 κ_2g(z_1,z_2) - κ_1^2 h(z_1) - κ_2^2 h(z_2), \] where $g:Ω\timesΩ\to\mathbb{R}$ is the regular part of a hydrodynamic Green function in $Ω$, $h:Ω\to\mathbb{R}$ is the Robin function: $h(z)=g(z,z)$, and $κ_1$, $κ_2$ are the vortex strengths. We prove the existence of infini…

The N-vortex Problem on a Riemann Sphere

This article investigates the dynamical behaviours of the $n$-vortex problem with vorticity $\mathbfΓ$ on a Riemann sphere $\mathbb{S}^2$ equipped with an arbitrary metric $g$. From perspectives of Riemannian geometry and symplectic geometry, we study the invariant orbits and prove that with some constraints on vorticity $\mathbfΓ$, the $n$-vortex problem possesses finitely many fixed points and infinitely many periodic orbits for generic $g$. Moreover, we verify the contact structure on hyper-surfaces of the vortex dipole, and exclude the existence of perverse symmetric orbits.

Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary

The paper deals with the existence of nonstationary collision-free periodic solutions of singular first order Hamiltonian systems of $N$-vortex type in a domain $Ω\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of \[ \dot{z}_j(t)=-i\nabla_{z_j} H_Ω\big(z(t)\big),\quad j=1,\dots,N, \tag{HS} \] where the Hamiltonian $H_Ω$ has the form \[ H_Ω(z_1,\dots,z_N) = -\sum_{{j,k=1}\over{j\ne k}}^N \frac{1}{2π}\log|z_j-z_k| -\sum_{j,k=1}^N g(z_j,z_k). \] The function $g:Ω\timesΩ\to\mathbb{R}$ is required to be of class $C^3$ and symmetric, the regular part of a hydrodynamic Green func…

Periodic solutions of the N-vortex Hamiltonian system in planar domains

We investigate the existence of collision-free nonconstant periodic solutions of the $N$-vortex problem in domains $Ω\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of the first order Hamiltonian system \[ \dot{z}_k(t)=-i\nabla_{z_k} H_Ω\big(z(t)\big),\quad k=1,\dots,N, \] where the Hamiltonian $H_Ω$ has the form \[ H_Ω(z_1,\dots,z_N) = \frac1{2π}\sum_{{j,k=1}\atop{j\ne k}}^N \log\frac1{|z_j-z_k|} - F(z). \] The function $F:Ω^N\to\mathbb{R}$ depends on the regular part of the hydrodynamic Green's function and is unbounded from above. The Hamiltonian is unbounded from above…

Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

The paper deals with singular first order Hamiltonian systems of the form \[ Γ_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in Ω\subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in\mathbb{R}^{2\times2}$ defines the standard symplectic structure in $\mathbb{R}^2$, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2π} \sum_{j\neq k=1}^N Γ_j Γ_k \log|z_j-z_k| - F(z). \] This is defined on the configuration space $\{(z_1,\ldots,z_N)\in Ω^{2N}:z_j\neq z_k\text{ for }j\neq k\}$ of $N$ different points in the domain $Ω\subset\mathbb{R}^2$. The function $F:Ω^N\to\…

双周期域中二维湍流的拓扑涡旋识别

Topological vortex identification for two-dimensional turbulent flows in doubly periodic domains

本文提出基于环面 Hamiltonian 流粒子轨道拓扑结构的分类理论，将全局轨道结构转换为部分循环有序根树及字符串表示，成功应用于二维湍流中相干涡旋结构的提取与统计特性研究。