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

TL;DR

The paper develops a renormalization-group framework for studying the inviscid limit and spontaneous stochasticity in multiscale dynamics on a fractal space-time lattice. It defines flow-map RG dynamics with a fixed-point attractor $\phi^{\infty}$ and analyzes universal first-order corrections via the leading eigenmode, revealing a period-doubling bifurcation that yields alternate inviscid limits. When the RG dynamics becomes chaotic, a stochastic RG operator acting on flow kernels $\Phi$ produces a fixed-point kernel $\Phi^{\infty}$, establishing spontaneously stochastic solutions whose limiting distributions are universal. The work also demonstrates a period-doubling bifurcation in the stochastic setting, yielding bifurcating attractors and parity-dependent, universal limiting statistics for the inviscid limit. Collectively, the results connect universality, bifurcation structure, and intermittency to an RG description of multiscale turbulence models, offering a tractable route to understand Eulerian spontaneous stochasticity and energy cascades.

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.

Figures (13)

• Figure 1: Fractal space-time lattice. The initial conditions $u_n(0) = a_n$ are shown as black dots, and the unknown states $u_n(t)$ are shown as white dots. The arrows indicate the functional dependence of states at different times: red arrows correspond to energy transfer, and blue dotted lines indicate the influence of adjacent scales on energy transfer.
• Figure 2: Energy transfer scheme: the fraction $f_n(t)$ of $u_n(t)$ is transferred to the smaller scale $n+1$ at the next time $t+\tau_{n+1}$, while the fraction $1-f_n(t)$ remains in the same scale. The value of $f_n(t)$ depends on the states at two adjacent scales, $u_n(t)$ and $u_{n+1}(t)$.
• Figure 3: Schematic representation of the RG operator transforming $\phi^{(N,\alpha)}$ to $\phi^{(N+1,\alpha)}$.
• Figure 4: Schematic representation of the relations (\ref{['eq3_GSex']}) and (\ref{['eq3_GSpsi']}).
• Figure 5: (a) Convergence (very accurate collapse) of the states $u(1) = \phi^{(N,\alpha)}(a)$ for large $N = 12,\ldots,15$ and regularization parameters $\alpha = 1/4$ and $3/4$. (b) Components of the corresponding eigenvector $\psi(a)$ computed by Eq. (\ref{['eq4_OmN']}) for the different parameters $(N,\alpha)$ (same as in the left panel); all the curves match very accurately. The inset shows the original differences $\Delta u = \phi^{(N+1,\alpha)}(a)-\phi^{(N,\alpha)}(a)$ before the rescaling.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof