论文标题

具有空间异质性SPDE的分叉和早期训练标志

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

论文作者

Bernuzzi, Paolo, Kühn, Christian

论文摘要

Bistability是在非线性科学中出现的许多系统的关键特性。例如,它出现在许多部分微分方程(PDE)中。对于标态双重反应扩张PDE,Biscable Case甚至在诸如Allee,Allee,Allen-Cahn,Chafee-Infante,Nagumo,Ginzburg-landau,$φ_4$,Schlögl,Schlögl,Stommel,Stommel,仅在结构上类似的Belbis-Bistable模型名称中命名等社区中的不同名称。一种关键机制,在参数变化下如何出现双叉性是干草叉分叉。特别是,在反应 - 扩散PDES上服用干草叉分叉法线形式是上述PDE家族中的另一个变体。更一般而言,考虑稳态和稳定性的该PDE类别的研究与由于参数引起的分叉有关,这对于确定性案例已经很好地理解了。对于随机PDE(SPDE)案例,情况不太了解,最近对此进行了研究。在本文中,我们概括并统一了SPDE分叉的最新结果。当我们在方程式上引入空间异质项并放松定义噪声的协方差操作员的假设时,我们的概括是直接由应用动机。对于这种空间异质性SPDE,我们证明了有限的Lyapunov指数分叉结果。此外,我们在上下文中扩展了预警迹象的理论,并解释说,协方差操作员警告标志与缺乏有限的Lyapunov统一性之间的普遍指数的作用。我们的结果伴随并通过数值模拟进行了交叉验证。

Bistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has take on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau, $Φ_4$, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. In this paper we generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain, the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

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