三维常微分方程中连接轨线的延拓：(II) 周期轨道到周期轨道的连接

高效地将数值继续与椭圆型PDEs的多重解方法结合

Efficient Gluing of Numerical Continuation and a Multiple Solution Method for Elliptic PDEs

本文研究了椭圆型PDEs的数值继续方法，通过pde2path软件开发了连接组件，利用变分结构寻找初始解，并通过极小极大算法解决初始化问题，揭示了对称破缺现象。

多尺度轨道的几何与数值持续性在非凸变分问题中

Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

研究非凸变分问题的多尺度结构，通过几何和数值方法分析周期解的性质，揭示参数对轨道行为的影响。

On transversal connecting orbits of Lagrangian systems in non-stationary force field: Newton-Kantorovich approach

We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to the action of a non-stationary force field with potential $U(q,t) = f(t)V(q)$. It is assumed that the factor $f(t)$ tends to $\infty$ as $t\to \pm\infty$ and vanishes at a unique point $t_{0}\in \mathbb{R}$. Let $X_{+}$, $X_{-}$ denote the sets of isolated critical points of $V(x)$ at which $U(x,t)$ as a function of $x$ attains its maximum for any fixed $t> t_{0}$ and $t<t_{0}$, respectively. Under nondegeneracy conditions on points of $X_{\pm}$ we apply Newton-Kantorovich type method to study …

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connect…

Methods of Continuation and their Implementation in the COCO Software Platform with Application to Delay Differential Equations

This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the COCO software package and its explicit support for community-driven development. To this end, the paper first formalizes the COCO construction paradigm for augmented continuation problems compatible with simultaneous analysis of implicitly defined manifolds of solutions to nonlinear equations and the corre…

具有四次色散的非线性薛定谔方程中的广义多振荡孤子

Generalized and multi-oscillation solitons in the Nonlinear Schrödinger Equation with quartic dispersion

本文研究具有二次和四次色散的广义非线性薛定谔方程中的孤子，通过将PDE化为四阶哈密顿ODE并利用同宿轨道数值方法，发现PtoP连接及其与EtoP连接组装的异宿循环可产生广义振荡孤子和多振荡孤子，并证明它们是截断同宿蛇形现象的重要组成部分。