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,…