RG analysis of spontaneous stochasticity on a fractal lattice: stability and bifurcations

TL;DR

This paper uses a renormalization group approach to analyze stability, bifurcations, and spontaneous stochasticity in models of turbulence on fractal lattices, revealing universal behaviors and bifurcation phenomena.

Contribution

It introduces a novel RG framework for understanding spontaneous stochasticity and bifurcations in turbulence models on fractal lattices, highlighting universality and stochastic attractors.

Findings

RG fixed points explain the universality of the inviscid limit.

Period-doubling bifurcations alter the nature of the inviscid limit.

Stochastic RG operators account for spontaneous stochasticity in chaotic regimes.

Abstract

In this paper, we study the stability and bifurcations of spontaneous stochasticity using an approach reminiscent of the Feigenbaum renormalization group (RG). We consider dynamical models on a self-similar space-time lattice as toy models for multiscale motion in hydrodynamic turbulence. Here an ill-posed ideal system is regularized at small scales and the vanishing regularization (inviscid) limit is considered. By relating the inviscid limit to the dynamics of the RG operator acting on the flow maps, we explain the existence and universality (regularization independence) of the limiting solutions as a consequence of the fixed-point RG attractor. Considering the local linearized dynamics, we show that the convergence to the inviscid limit is governed by the universal RG eigenmode. We also demonstrate that the RG attractor undergoes a period-doubling bifurcation with parameter variation, thereby changing the nature of the inviscid limit. In the case of chaotic RG dynamics, we introduce the stochastic RG operator acting on Markov kernels. Then the RG attractor becomes stochastic, which explains the existence and universality of spontaneously stochastic solutions in the limit of vanishing noise. We study a linearized structure (RG eigenmode) of the stochastic RG attractor and its period-doubling bifurcation. Viewed as prototypes of Eulerian spontaneous stochasticity, our models explain its mechanism, universality and potential diversity.