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Mean field coupled dynamical systems: Bifurcations and phase transitions

Wael Bahsoun, Carlangelo Liverani

TL;DR

The article develops a rigorous, abstract bifurcation framework for infinite-dimensional mean-field coupled chaotic maps, recasting invariant-measure dynamics through a nonlinear transfer operator on Banach scales and an implicit-function analysis of the fixed-point equation $F(\nu,h)=0$. It proves local and global bifurcation results, delineating when multiple invariant (SRB/physical) measures can arise as the coupling strength $\nu$ varies, and provides criteria for physicality based on spectral data of the linearized operator ${\mathcal D}_h$. The paper then applies the theory to a concrete model of globally coupled Anosov diffeomorphisms, demonstrating a phase transition: for small coupling there is a unique physical state, while for larger coupling there are multiple physical states, a phenomenon that only manifests in the infinite-dimensional limit. This work bridges bifurcation theory, nonlinear transfer-operator methods, and statistical-mechanics notions of physical observables, highlighting how phase transitions emerge in mean-field dynamical systems in the thermodynamic limit.

Abstract

We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses that are tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled expanding maps to globally coupled Axiom A diffeomorphisms. To illustrate the range of applicability, we analyze an explicit example consisting of globally coupled Anosov diffeomorphisms. For such an example, we classify all the invariant measures as the coupling strength varies; we show which invariant measures are physical, and we prove that the existence of multiple invariant physical measures is a purely infinite dimensional phenomenon, i.e., the model exhibits phase transitions in the sense of statistical mechanics.

Mean field coupled dynamical systems: Bifurcations and phase transitions

TL;DR

The article develops a rigorous, abstract bifurcation framework for infinite-dimensional mean-field coupled chaotic maps, recasting invariant-measure dynamics through a nonlinear transfer operator on Banach scales and an implicit-function analysis of the fixed-point equation . It proves local and global bifurcation results, delineating when multiple invariant (SRB/physical) measures can arise as the coupling strength varies, and provides criteria for physicality based on spectral data of the linearized operator . The paper then applies the theory to a concrete model of globally coupled Anosov diffeomorphisms, demonstrating a phase transition: for small coupling there is a unique physical state, while for larger coupling there are multiple physical states, a phenomenon that only manifests in the infinite-dimensional limit. This work bridges bifurcation theory, nonlinear transfer-operator methods, and statistical-mechanics notions of physical observables, highlighting how phase transitions emerge in mean-field dynamical systems in the thermodynamic limit.

Abstract

We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses that are tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled expanding maps to globally coupled Axiom A diffeomorphisms. To illustrate the range of applicability, we analyze an explicit example consisting of globally coupled Anosov diffeomorphisms. For such an example, we classify all the invariant measures as the coupling strength varies; we show which invariant measures are physical, and we prove that the existence of multiple invariant physical measures is a purely infinite dimensional phenomenon, i.e., the model exhibits phase transitions in the sense of statistical mechanics.
Paper Structure (12 sections, 29 theorems, 226 equations, 3 figures)

This paper contains 12 sections, 29 theorems, 226 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The model and an abstract bifurcation theory
  3. Regularity properties
  4. Properties of the functions $H$ and $F$
  5. Characterization of invariant measures
  6. Local existence of invariant measures
  7. Global existence of invariant measures
  8. Local stability and physical measures
  9. Example
  10. The bifurcation diagram
  11. Phase transition
  12. A quantitative Implicit Function Theorem

Key Result

Lemma 2.8

Under assumptions (Aass:6) and (Aass:4), for each $\{S_j\}_{j=1}^{\bar{n}}\subset {\mathcal{T}}$ we have ${\mathcal{L}}_{S_{\bar{n}}}\cdots{\mathcal{L}}_{S_1}{\mathbb W}\subset {\mathbb W}$. In addition, for each $S\in {\mathcal{T}}$, $\sigma_{{\mathcal{B}}_1}({\mathcal{L}}_S)\subset \{1\}\cup\{z\in

Figures (3)

  • Figure 1: A possible bifurcation diagram in Theorem \ref{['thm:implicitg']}. The red curve, $(G(\tau),{\mathbbm h}(\tau))$, is the set of the invariant measures $h_\nu$ (which satisfy the relation $h_{G(\tau)}={\mathbbm h}(\tau)$).
  • Figure 2: Another possible scenario for the bifurcation diagram in Theorem \ref{['thm:implicitg']} in which two curve of solutions exist.
  • Figure 3: Diagram corresponding to Proposition \ref{['prop:exsols']}. The red curve is the set of the invariant measures $h_\nu$.

Theorems & Definitions (74)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Theorem 2.9
  • ...and 64 more