ikeda_DDE2

池田遅延微分方程式2
Ikeda delay differential equation2

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 sin2( x(t - τ) - C ) - B x(t)

case 1

parameters: A = 20.0, B = 1.0, C = π/4, τ = 5, DIM = 100
The step size of the fourth-order Runge-Kutta method: τ/DIM = 0.05

Lyapunov exponents (log with base-e): 0.2112, 0.1956, 0.1783, 0.1631, 0.1523 (the first five exponents)
Lyapunov dimension: 42.2406

case 2

parameters: A = 20.0, B = 1.0, C = π/4, τ = 0.478, DIM = 100
The step size of the fourth-order Runge-Kutta method: τ/DIM = 0.00478
Lyapunov exponents (log with base-e): 0.5840, 0.0067, -1.7263, -2.8788, -3.4444 (the first five exponents)
Lyapunov dimension: 2.3422