arXiv subjects

Paolo Bernuzzi

Publications and source records attributed to Paolo Bernuzzi.

7 recordsLinked to original sources

arXiv2026

Impact of spinning on the early-warning signs in non-Markovian stochastic systems

We construct early-warning signals for impending critical transitions in non-Markovian systems. We analyze stochastic forcings such as fractional Brownian motion, fractional Ornstein-Uhlenbeck processes and red noise in fast-slow systems exhibiting such transitions. We show that the effectiveness of indicators such as autocovariance, autocorrelation, and spectral density depends on several properties of the underlying system. In particular, we compare the influence of the Hurst index and the bifurcation type. We prove that the rotatory dynamics associated with a Hopf bifurcation substantially alters the scaling laws of these observables. Finally, we provide practical guidelines for implementing these signals and validate them on both theoretical and applied models.

math.PR↗

arXiv2025

Critical Slowing Down in Bifurcating Stochastic Partial Differential Equations with Red Noise

The phenomenon of critical slowing down (CSD) has played a key role in the search for reliable precursors of catastrophic regime shifts. This is caused by its presence in a generic class of bifurcating dynamical systems. Simple time-series statistics such as variance or autocorrelation can be taken as proxies for the phenomenon, making their increase a useful early warning signal (EWS) for catastrophic regime shifts. However, the modelling basis justifying the use of these EWSs is usually a finite-dimensional stochastic ordinary differential equation, where a mathematical proof for the aptness is possible. Only recently has the phenomenon of CSD been proven to exist in infinite-dimensional stochastic partial differential equations (SPDEs), which are more appropriate to model real-world spatial systems. In this context, we provide an essential extension of the results for SPDEs under a specific noise forcing, often referred to as red noise. This type of time-correlated noise is omnipresent in many physical systems, such as climate and ecology. We approach the question with a mathematical proof and a numerical analysis for the linearised problem. We find that also under red noise forcing, the aptness of EWSs persists, supporting their employment in a wide range of applications. However, we also find that false or muted warnings are possible if the noise correlations are non-stationary. We thereby extend a previously known complication with respect to red noise and EWSs from finite-dimensional dynamics to the more complex and realistic setting of SPDEs.

math.PR↗

arXiv2025

Probability of Transition to Turbulence in a Reduced Stochastic Model of Pipe Flow

We study the phenomenon of turbulence initiation in pipe flow under different noise structures by estimating the probability of initiating metastable transitions. We establish lower bounds on turbulence transition probabilities using linearized models with multiplicative noise near the laminar state. First, we consider the case of stochastic perturbations by It\^o white noise; then, through the Stratonovich interpretation, we extend the analysis to noise types such as white and red noise in time. Our findings demonstrate the viability of detecting the onset of turbulence as rare events under diverse noise assumptions. The results also contribute to applied SPDE theory and offer valuable methodologies for understanding turbulence across application areas.

math.PR↗

arXiv2024

Large Deviation Minimisers for Stochastic Partial Differential Equations with Degenerate Noise

Noise-induced transitions between multistable states happen in a multitude of systems, such as species extinction in biology, protein folding, or tipping points in climate science. Large deviation theory is the rigorous language to describe such transitions for non-equilibrium systems in the small noise limit. At its core, it requires the computation of the most likely transition pathway, solution to a PDE constrained optimization problem. Standard methods struggle to compute the minimiser in the particular coexistence of (1) multistability, i.e. coexistence of multiple long-lived states, and (2) degenerate noise, i.e. stochastic forcing acting only on a small subset of the system's degrees of freedom. In this paper, we demonstrate how to adapt existing methods to compute the large deviation minimiser in this setting by combining ideas from optimal control, large deviation theory, and numerical optimisation. We show the efficiency of the introduced method in various applications in biology, medicine, and fluid dynamics, including the transition to turbulence in subcritical pipe flow.

math.PR↗

arXiv2024

Warning Signs for Boundary Noise and their Application to an Ocean Boussinesq Model

In this paper, we construct and discuss early-warning signs of the approach of a parameter to a deterministic bifurcation on a stochastic partial differential equation (SPDE) model with Gaussian white-noise on the boundary. We specifically focus on the qualitative behaviour of the time-asymptotic autocovariance and autocorrelation of the solutions of the linearized system. We also discuss the reliability of the tools from an analytic perspective and through various examples. Among those, the application of the early-warning signs for an ocean Boussinesq model is explored through numerical simulations. The analytic results obtained expand on previous work and present valuable early-warning signs for various applications.

math.PR↗

arXiv2023

Early-Warning Signs for SPDEs with Continuous Spectrum

In this work, we study early-warning signs for stochastic partial differential equations (SPDEs), where the linearization around a steady state has continuous spectrum. The studied warning sign takes the form of qualitative changes in the variance as a deterministic bifurcation threshold is approached via parameter variation. Specifically, we focus on the scaling law of the variance near the transition. Since we are dealing here, in contrast to previous studies, with the case of continuous spectrum and quantitative scaling laws, it is natural to start with linearizations that are multiplication operators defined by analytic functions. For a one-dimensional spatial domain we obtain precise rates of divergence. In the case of the two- and three-dimensional domains an upper bound to the rate of the early-warning sign is proven. These results are cross-validated by numerical simulations. Our theory can be generically useful for several applications, where stochastic and spatial aspects are important in combination with continuous spectrum bifurcations.

math.PR↗

arXiv2022

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

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.

math.SP↗