ローレンツ方程式
Lorenz equations

dx/dt = σ (y - x)

dy/dt = -x z + R x - y

dz/dt = x y - B z


parameters: σ = 10, R = 28, B = 8/3 (Lorenz used these values)
The step size of the fourth-order Runge-Kutta method: 0.001

Lyapunov exponents (log with base-e): 0.9050, -0.0001, -14.5715

Lyapunov dimension: 2.0621

The sum of the Lyapunov exponents is theoretically the same as -σ - 1 - B.


Reference

Deterministic nonperiodic flow

Edward Norton Lorenz

Journal of Atmospheric Sciences, 20, 130-141 (1963)

DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2


Computing the Lyapunov spectrum of a dynamical system from an observed time series

Reggie Brown, Paul Bryant, and Henry Don Isaac Abarbanel

Physical Review A 43, 2787-2806 (1991)

DOI: 10.1103/PhysRevA.43.2787

programme

makefile

lorenz_lyap.h

lorenz_lyap.c

solve_lorenz_eqs.c

gram_schmidt_orth.c


lorenz_lyap.c contains the main function.

QR分解を用いてローレンツ方程式のリアプノフスペクトルを推定するプログラムは、こちら
A programme for estimating Lyapunov spectrum of the Lorenz equations using the QR decomposition is available here.