Criticality Part 2: Bifurcations, Universality, and Self-Organization

Introduction

• Recap of Part 1: tipping point vs. continuous criticality
• What’s missing: the geometric picture, why different systems share exponents, systems that find criticality on their own
• Goal: connect phenomenological classification to deeper mathematical frameworks

Bifurcations: The Deterministic Skeleton

Phase Portraits and Fixed Points

• State space, trajectories, attractors
• Fixed point stability: linearization and eigenvalues
• Basins of attraction: which initial conditions lead where

Catalog of Local Bifurcations

Saddle-Node Bifurcation

• Normal form: $\dot{x} = r + x^2$
• Two fixed points collide and annihilate
• The canonical “tipping point”—no smooth return
• Examples: extinction thresholds, financial crises, regime shifts in ecosystems

Transcritical Bifurcation

• Normal form: $\dot{x} = rx - x^2$
• Two fixed points exchange stability
• Connection to exponential growth model from Part 1

Pitchfork Bifurcation

• Supercritical: $\dot{x} = rx - x^3$ (smooth symmetry breaking)
• Subcritical: $\dot{x} = rx + x^3$ (discontinuous jump)
• Connection to continuous vs. tipping point criticality

Hopf Bifurcation

• Fixed point gives way to limit cycle
• Supercritical vs. subcritical variants
• Examples: oscillations in predator-prey systems, electrical circuits

Global Bifurcations

• Homoclinic and heteroclinic bifurcations
• Why local analysis isn’t always enough
• Catastrophe theory: Thom’s classification

Interactive: Bifurcation Explorer

• Phase portrait that deforms as control parameter changes
• Visualize fixed point creation/destruction/stability exchange
• Show corresponding bifurcation diagram

From Bifurcations to Statistical Mechanics

The Role of Many Degrees of Freedom

• Single ODE → lattice of coupled units
• Fluctuations and noise enter naturally
• Mean-field theory: averaging over neighbors recovers bifurcation-like equations

Landau Theory of Phase Transitions

• Free energy as a function of order parameter
• Equilibrium: minimize free energy
• Taylor expansion near transition: $F(m) = a(T)m^2 + bm^4 + \ldots$
• Connection to pitchfork bifurcation

Order Parameter Behavior Near Criticality

• Power-law scaling: $m \sim |T - T_c|^\beta$
• Critical exponent $\beta$ characterizes the transition
• Mean-field prediction vs. reality: fluctuations matter in low dimensions

Correlation Length and Susceptibility

• Correlation length $\xi$: how far do fluctuations propagate?
• $\xi \sim |T - T_c|^{-\nu}$ diverges at criticality
• Susceptibility $\chi \sim |T - T_c|^{-\gamma}$: response to perturbations diverges
• Critical slowing down: relaxation time also diverges

Universality: Why Details Don’t Matter

The Puzzle

• Liquid-gas transition in water and magnetization in iron follow same power laws
• Microscopic physics completely different
• How can this be?

What Determines Universality Class

• Dimensionality of space
• Symmetry of order parameter (scalar, vector, etc.)
• Range of interactions (short vs. long range)
• What doesn’t matter: lattice structure, precise interaction strengths, chemical details

The Renormalization Group Idea

Coarse-Graining

• Block spins: average over small regions
• Ask: what effective Hamiltonian describes the coarse-grained system?
• Iterate: zoom out repeatedly

RG Flow and Fixed Points

• Parameters flow under coarse-graining
• Fixed points of the flow = scale-invariant systems = critical points
• Behavior near fixed point determines critical exponents

Intuitive Picture

• At criticality, zooming out doesn’t change statistical properties
• The system “looks the same” at all scales
• This self-similarity is the origin of power laws

Scaling Relations

• Critical exponents aren’t independent
• Relations like $\alpha + 2\beta + \gamma = 2$ (Rushbrooke)
• Follow from self-consistency of RG picture
• Reduces number of independent exponents

Table of Universality Classes

Class	Examples	$\beta$	$\nu$	$\gamma$
2D Ising	Uniaxial magnets, lattice gas	1/8	1	7/4
3D Ising	Liquid-gas, binary mixtures	~0.326	~0.630	~1.237
Mean-field	High dimensions, long-range	1/2	1/2	1
2D Percolation	Porous media, networks	5/36	4/3	43/18
3D Percolation	Composite materials	~0.41	~0.88	~1.80

Fractals at Criticality

Why Fractals Emerge

• No characteristic length scale at $p_c$ or $T_c$
• Structures must be self-similar
• Fractal geometry is the natural description

Fractal Dimension

• Definition: $M(L) \sim L^{d_f}$ where $d_f$ is fractal dimension
• For percolation cluster at $p_c$: $d_f < d$ (spatial dimension)
• 2D percolation: $d_f = 91/48 \approx 1.896$
• The cluster fills space “incompletely”

Fractal Dimension as Critical Exponent

• Related to other exponents through scaling relations
• $d_f = d - \beta/\nu$ connects geometry to thermodynamics
• Measuring $d_f$ gives information about universality class

Examples of Critical Fractals

• Percolation clusters at threshold
• Ising domain boundaries at $T_c$
• Coastlines and river networks (approximately)
• Diffusion-limited aggregation

Interactive: Box-Counting on Critical Cluster

• Generate percolation cluster at $p = p_c$
• Overlay boxes of decreasing size
• Plot $\log N$ vs. $\log (1/\epsilon)$
• Slope gives fractal dimension

Self-Organized Criticality

The Tuning Problem

• In standard criticality: must tune control parameter precisely to $p_c$
• But power laws appear everywhere in nature
• Who is doing the tuning?

SOC: Criticality Without Tuning

• Some systems naturally evolve toward critical state
• Separation of timescales is key: slow drive, fast relaxation
• System “self-organizes” to the boundary between stability and instability

The BTW Sandpile Model

Rules

• 2D lattice, each site has “sand grains” $z_{ij}$
• Slowly add grains at random sites
• If $z_{ij} \geq 4$, site topples: loses 4 grains, neighbors each gain 1
• Toppling can trigger neighbors → avalanche
• Grains fall off at boundaries

Emergent Behavior

• System reaches statistical steady state
• Avalanche size distribution follows power law: $P(s) \sim s^{-\tau}$
• No characteristic avalanche size
• Always poised at criticality

Other Examples of SOC

• Earthquakes: Gutenberg-Richter law $P(E) \sim E^{-b}$
• Forest fires: burned area distribution
• Mass extinctions: species loss events
• Neural avalanches in brain activity
• Solar flares: energy release distribution

Controversies and Limitations

• Is true SOC common, or are power laws often misidentified?
• Many claimed power laws don’t survive rigorous statistical testing
• Alternative mechanisms: finite-size effects, superposition, tuning to different points
• SOC requires specific conditions: slow drive, fast relaxation, no characteristic scale in driving

Interactive: Sandpile Simulation

• Add grains one by one
• Visualize avalanches as they propagate
• Plot avalanche size histogram (log-log)
• Show approach to power-law distribution

Synthesis: A Unified View of Criticality

The Common Thread

• Criticality = diverging correlation length = loss of characteristic scale
• Power laws, fractals, and scale invariance are manifestations of same phenomenon
• Universality: only symmetry and dimension matter near the fixed point

Hierarchy of Descriptions

Bifurcation Theory (deterministic, low-dimensional)
         ↓ add noise, many degrees of freedom
Mean-Field / Landau Theory (averaged, approximate)
         ↓ include fluctuations properly
Renormalization Group (full critical behavior)
         ↓ special dynamics
Self-Organized Criticality (criticality as attractor)

Mapping the Landscape

Framework	Tipping Point	Continuous Transition
Bifurcations	Saddle-node, subcritical	Supercritical pitchfork, transcritical
Phase transitions	First-order	Second-order (continuous)
Order parameter	Discontinuous jump	Power-law approach to zero
Correlation length	Finite	Diverges
Hysteresis	Often present	Absent
Critical exponents	Not applicable	Universal values

Early Warning Signals

• Approaching tipping points: critical slowing down
• Variance and autocorrelation increase before transition
• Flickering between states
• Potential for prediction and intervention

Open Questions

• Criticality in non-equilibrium systems: active matter, driven systems
• Quantum criticality: phase transitions at zero temperature
• Criticality and computation: is the brain critical? Why might this be advantageous?
• Machine learning on critical systems: can neural networks learn critical exponents?

Conclusion

• Criticality connects dynamical systems, statistical mechanics, and geometry
• The same mathematical structures appear across physics, biology, and social systems
• Understanding criticality gives tools for prediction, control, and recognizing universal patterns
• Much remains unknown, especially in non-equilibrium and living systems

Sources

• Nonlinear Dynamics and Chaos by Steven H. Strogatz
• Statistical Mechanics by James Sethna (especially Chapter 12 on RG)
• Scale Invariance and Beyond edited by Dubrulle, Graner, Sornette
• How Nature Works by Per Bak (original SOC book)
• Critical Phenomena in Natural Sciences by Didier Sornette
• Clauset, Shalizi, Newman: “Power-Law Distributions in Empirical Data” (arXiv:0706.1062)