Embedded Files
マッキー・グラス方程式
Consider a delay differential equation of the form
dx/dt = F[x(t), x(t-τ)]
where τ (TAU) is a constant delay time. This equation is infinite-dimensional in the sense that a continuum of initial conditions over the interval -τ < t < 0 is required to specify the behaviour. That is, the number of variables in the equation is also infinite. Hence, DIM we set should be larger values and I use DIM larger than 100. It should be noted that, for example, if DIM = 1000, it will need extremely long time. The integration step DELTA_T should be equal to TAU/DIM.
dx/dt = A x(t - 1)/( 1 + x(t - τ)C ) - B x(t)
parameters: A = 0.2, B = 0.1, C = 10, τ = 17, DIM = 100
The step size of the fourth-order Runge-Kutta method: τ/DIM = 0.17
Lyapunov exponents (log with base-e): 0.0056, -0.0002, -0.0394, -0.0552, -0.0694 (the first five exponents)
Lyapunov dimension: 2.1374
Reference
Oscillation and Chaos in Physiological Control Systems
Michael Charles Mackey and Leon Glass
Science 197, 4300, 287–289 (1977)
Chaotic attractors of an infinite-dimensional dynamical system
Jamse Doyne Farmer
Physica D 4, 366-393 (1982)
DOI: 10.1016/0167-2789(82)90042-2
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)
programme
mackey-glass_DDE_lyap.c contains the main function.
Page updated
Google Sites
Report abuse