[Paper Review] Some fractal aspects of Self-Organized Criticality

Bruno Cessac|arXiv (Cornell University)|Sep 2, 2004

Complex Systems and Time Series Analysis5 references3 citations

TL;DR

This paper formulates the Zhang model of Self-Organized Criticality as a piecewise hyperbolic dynamical system and iterated function system, linking the fractal geometry of the invariant measure to avalanche statistics. It derives a lower bound on avalanche size scaling logarithmically with system size, implying a critical exponent τ < 2, and connects Lyapunov exponents, Hausdorff dimension, and energy transport via the Ledrappier-Young formula.

ABSTRACT

The concept of Self-Organized Criticality (SOC) was proposed in an attempt to explain the widespread appearance of power-law in nature. It describes a mechanism in which a system reaches spontaneously a state where the characteristic events (avalanches) are distributed according to a power law. We present a dynamical systems approach to Self-Organized Criticality where the dynamics is described either in terms of Iterated Function Systems, or as a piecewise hyperbolic dynamical system of skew-product type. Some results linking the structure of the attractor and some characteristic properties of avalanches are discussed.

Motivation & Objective

• To establish a rigorous dynamical systems framework for Self-Organized Criticality (SOC) models, particularly the Zhang model, to overcome limitations of purely numerical or heuristic approaches.
• To investigate the connection between the fractal structure of the invariant measure (attractor) and the power-law statistics of avalanches in SOC systems.
• To explore how geometric and ergodic properties—such as Hausdorff dimension, Lyapunov exponents, and multifractal spectra—determine the scaling behavior of avalanche observables.
• To derive analytical bounds on the critical exponent τ of the avalanche size distribution using dynamical systems tools like the Ledrappier-Young formula.
• To clarify the role of energy transport mechanisms in SOC by relating them to the singular values and eigenmodes of the relaxation map’s Jacobian.

Proposed method

• Modeling the Zhang SOC system as a piecewise hyperbolic dynamical system of skew-product type, with state evolution governed by a relaxation map F(X) = X + αΔ[Z(X)∗X], where Δ is the discrete Laplacian with zero boundary conditions.
• Representing the dynamics using iterated function systems (IFS), where each avalanche corresponds to a sequence of maps acting on phase space domains M_(i,j), forming a partition of the configuration space M.
• Analyzing the invariant measure μ_L via the Ledrappier-Young formula, relating the sum of positive Lyapunov exponents, partial Hausdorff dimensions σ_L(i), and Kolmogorov-Sinai entropy.
• Using thermodynamic formalism for non-conformal maps to compute the multifractal spectrum from the singular values of the product of Jacobian matrices along trajectories.
• Relating the singular values of the tangent map D F_X to the eigenmodes of energy propagation, which define a hierarchy of characteristic times for energy transport (singular, anomalous, normal regimes).
• Applying backward dynamics and preimage counting to derive the entropy term log(N) − log(J_N), which enters the Ledrappier-Young equation and enables bounds on avalanche size.

Experimental results

Research questions

• RQ1 How does the fractal structure of the invariant measure in the Zhang model relate to the power-law distribution of avalanche sizes?
• RQ2 What is the relationship between the Lyapunov exponents, partial Hausdorff dimensions, and the critical exponent τ of the avalanche size distribution?
• RQ3 How does the multifractal spectrum of the invariant measure reflect the anisotropic and multiscale nature of energy transport in the system?
• RQ4 In what way does the critical energy E_c influence the Hausdorff dimension of the attractor, and is this dependence continuous?
• RQ5 Can the dynamics of energy propagation in the Zhang model be characterized through the eigenmodes of the relaxation map’s Jacobian matrix?

Key findings

• The Hausdorff dimension of the invariant measure μ_L is conjectured to be piecewise continuous and monotonically increasing with E_c on intervals where the map structure remains unchanged.
• A lower bound on the average avalanche size is derived as ⟨s⟩_L ≥ (α / |log(ε)|) log(N), with strict equality if all partial Hausdorff dimensions are 1, implying that singular measures lead to super-logarithmic divergence of ⟨s⟩_L.
• The critical exponent τ of the avalanche size distribution satisfies τ < 2 when the invariant measure is singular, which follows from the divergence of ⟨s⟩_L under such conditions.
• The multifractal spectrum is computed via a sub-additive thermodynamic formalism based on the singular values of the Jacobian matrix product along trajectories, linking it to the energy transport hierarchy.
• The eigenmodes of the tangent map D F_X correspond to characteristic time scales of energy propagation: singular on short scales, anomalous at intermediate scales, and normal on long scales.
• The Ledrappier-Young formula links the sum of positive Lyapunov exponents and partial Hausdorff dimensions to the backward entropy, enabling a rigorous connection between geometric measure properties and statistical observables.