A Lipschitz version of the $λ$-Lemma and a characterization of homoclinic and heteroclinic orbits

Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations

In this paper, we consider the scalar reaction-diffusion equations $\partial_t u=Δu + f(x,u,\nabla u)$ on a bounded domain $Ω\subset\mathbb{R}^d$ of class $C^2$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unpr…

可逆系统中从异宿环到同宿蛇形现象：通过计算机辅助证明的严格强迫

From heteroclinic loops to homoclinic snaking in reversible systems: rigorous forcing through computer-assisted proofs

本文利用计算机辅助证明在时间可逆系统中构造平衡点与周期轨道之间的异宿连接参数化环，从而严格证明了 Swift-Hohenberg 和 Gray-Scott 问题中的同宿蛇形现象，展示了计算机辅助方法在分析非线性系统全局动力学中的有效性。

Slow localized patterns in singularly perturbed 2-component reaction-diffusion equations

Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts -- in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system -- alternated by fast excursions -- in which the solution jumps from one slow manifold to another, or back to the original slow manifold. In this paper we consider the existence and stability of stationary and traveling slow localized patterns that do not exhibit such jumps, i.e. that are completely embedded in a slow manifold of the singularly perturbed spatial dynamical system. The…

Homoclinic Tubes in Discrete Nonlinear Schroedinger Equation Under Hamiltonian Perturbations

In this paper, we study the discrete cubic nonlinear Schroedinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Backlund-Darboux transformations are utilized to generate heteroclinic orbits and Melnikov vectors. Then we give coordinate-expressions for persistent invariant manifolds and Fenichel fibers for the perturbed lattice. Finally based upon the above machinery, existence of codimension 2 transversal homoclinic tubes is established through a Melnikov type ca…

非自治 Minkowski 曲率方程的同宿解与异宿解

Homoclinic and heteroclinic solutions for non-autonomous Minkowski-curvature equations

本文利用相平面分析研究非自治 Minkowski 曲率方程，证明异宿解与具有唯一单调性变化点的同宿解的存在性，并分析其渐近行为。

Chaos in PDEs and Lax Pairs of Euler Equations

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1). transversal homoclinic orbits, (2). Silnikov homoclinic orbits. Around the transversal homoclinic orbits in infinite dimensional autonomous systems, the author was able to prove the existence of chaos through a shadowing lemma. Around the Silnikov homoclinic orbits, the author was able to prove the existence of chaos through a horseshoe construction. Very recently, there has be…