Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

Stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem

In this work, we present results on stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem. We show that this nonlocal version of the well-known Chafee-Infante equation bares some resemblance with the local version. However, its nonlocal characteristc requires a fine analysis of the spectrum of the associated linear operators, a lot more ellaborated than the local case. The saddle point property of equilibria is shown to hold for this quasilinear model.

Bifurcation and hyperbolicity for a nonlocal quasilinear parabolic problem

In this article, we study a one-dimensional nonlocal quasilinear problem of the form $u_t=a(\Vert u_x\Vert^2)u_{xx}+νf(u)$, with Dirichlet boundary conditions on the interval $[0,π]$, where $0<m\leq a(s)\leq M$ for all $s\in \mathbb{R}^+$ and $f$ satisfies suitable conditions. We give a complete characterization of the bifurcations and of the hyperbolicity of the corresponding equilibria. With respect to the bifurcations we extend the existing result when the function $a(\cdot)$ is non-decreasing to the case of general smooth nonlocal diffusion functions showing that bifurcations may be pitchf…

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded \textit{random hyperbolic solution} for the associated nonlinear nonautonomous…

Generic Morse-Smale property for the parabolic equation on the circle

In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showi…

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…

Sierpinski 垫片上的 Kuramoto 模型 II：扭曲态

The Kuramoto model on the Sierpinski Gasket II: Twisted states

本研究分析了Sierpinski垫片图上Kuramoto模型的稳定平衡点，证明了每个同伦类存在唯一稳定平衡，推广了经典扭曲态，并揭示了自相似组织与网络动力学的关联。