Interdisciplinary study of many-body systems whose aggregate behaviour emerges from local interactions and cannot be reduced to the sum of their parts. Canonical models: Kuramoto 1975 N-oscillator synchronisation with all-to-all…
complex-systems
Kuramoto 1975 model of N coupled phase-oscillators
Kuramoto 1975 (Int. Symp. Math. Prob. Theor. Phys.) dynamics of N coupled phase oscillators with natural frequencies ω_i drawn from density…
May 1976 logistic map x_{n+1}=r·x_n·(1−x_n): period-doubling cascade
May 1976 (Nature 261:459) popularised the logistic difference equation as the minimal model of non-linear dynamics and chaos. Fixed…
Broadbent-Hammersley percolation and Kesten's p_c=1/2 for Z² bond
Broadbent-Hammersley 1957 (Proc. Camb. Phil. Soc. 53:629) introduced percolation as the complementary problem to diffusion: on a lattice,…
Kuramoto critical coupling K_c = 2γ for Lorentzian natural-frequency distribution
Closed-form critical coupling for the Kuramoto model with Lorentzian natural-frequency distribution. Kuramoto's self-consistency equation…
First period-doubling bifurcation of logistic map at r=3 (|f'(x*)| = 1)
Closed-form bifurcation analysis. For the logistic map f(x) = rx(1−x), non-zero fixed point x* = 1 − 1/r is stable whenever |f'(x*)| < 1. …
Kesten 1980: bond percolation on Z² has p_c = 1/2 exactly
Kesten 1980 (Comm. Math. Phys. 74:41) rigorous proof via self-duality. The dual of the square lattice is itself Z² (shifted by (1/2,…
Lorenz attractor: (sigma, rho, beta) = (10, 28, 8/3); Hopf bifurcation at rho_H ~ 24.74; fixed points C+/-
Lorenz attractor / deterministic-chaos framework. Setup: Lorenz 1963 three-variable truncation of Saltzman's Rayleigh-Benard convection…
Feigenbaum universality: period-doubling ratio (r_{n+1}-r_n)/(r_{n+2}-r_{n+1}) -> delta = 4.66920...
Feigenbaum period-doubling universality framework. Setup: for a smooth unimodal 1D map f_r(x) (e.g. logistic x -> r*x*(1-x), sine x ->…
Poincare-Bendixson theorem: bounded 2D autonomous ODE trajectories have periodic or fixed-point omega-limit
Poincare-Bendixson dichotomy framework. Setup: planar (2D) autonomous smooth ODE dx/dt = f(x, y), dy/dt = g(x, y) with forward orbit…
Theorem: Lorenz fixed points C+/- at rho=28, beta=8/3 have x = sqrt(72) = 6*sqrt(2), z = 27
Theorem (Lorenz fixed-point canonical coordinates): at the classical Lorenz parameters (sigma=10, rho=28, beta=8/3), the non-trivial fixed…
Theorem: Feigenbaum delta ~ 4.669201609 with rel. error < 1e-8 vs 4.66920160910299
Theorem (Feigenbaum delta numerical-value canonical): the Feigenbaum period-doubling constant delta has reference value delta_ref =…
Theorem: van der Pol (mu>0) has unique attracting limit cycle in R^2 via Poincare-Bendixson
Theorem (van der Pol unique-limit-cycle via Poincare-Bendixson): the 2D phase-space (x, y = dx/dt) of the van der Pol oscillator d^2x/dt^2…
Self-organized criticality (Bak-Tang-Wiesenfeld)
BTW 1987 sandpile model: dissipative-driven systems naturally evolve toward critical state without parameter tuning; power-law avalanches;…
Emergence (strong vs weak)
Bedau 1997: weak emergence = high-level pattern reducible-in-principle to micro; strong emergence = high-level causal-power irreducible;…
Scale-free networks (Barabasi-Albert)
Barabasi-Albert 1999: preferential-attachment generates power-law degree distributions P(k) ~ k^-3; ubiquitous in WWW, citation networks,…
Small-world (Watts-Strogatz 1998)
Watts-Strogatz 1998: rewiring lattice with probability p produces small-world network with high clustering + low average path-length;…
Kuramoto model (synchronization)
Kuramoto 1975: N coupled oscillators with frequency-distribution g(omega); above critical coupling K_c, partial synchronization emerges;…
Agent-based modeling (Axelrod)
Axelrod 1984 / 1997: bottom-up simulation of agents with simple rules; reproduces complex emergent patterns (cooperation evolution /…
Reentrant phase transition
Modern reentrant: liquid-crystal-Cladis 1975 + spin-glass + Aslam 2018; nonmonotonic phase-boundaries.
Kuramoto sync (1975)
Y Kuramoto 1975 + Strogatz 2000 sync-review; modern brain-network + power-grid + Josephson-junction.
Flocking (Vicsek 1995)
T Vicsek 1995 self-propelled-particle flocking; modern bird-flocking-Cavagna 2008 + active-matter MIPS Cates 2015.
Cascade (Bikhchandani 1992)
Bikhchandani-Hirshleifer-Welch 1992 information-cascades; modern social-network virality + financial herding.
Fractal (Mandelbrot 1982)
B Mandelbrot 1982 'The Fractal Geometry of Nature'; modern fractal-dimension D_q + multifractal-analysis applications.
Emergence (Anderson 1972)
P Anderson 1972 'More is Different'; modern Bedau 1997 weak-emergence; basis of complexity-science philosophy.