ikeda_DDE1

池田遅延微分方程式1
Ikeda delay differential equation1

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 sin( x(t - τ) - C ) - B x(t)

case 1

parameters: A = 6.0, B = 1.0, C = 0, τ = 1, DIM = 100
The step size of the fourth-order Runge-Kutta method: τ/DIM = 0.01

Lyapunov exponents (log with base-e): 0.4375, -0.0060, -0.5017, -1.0210, -1.3043 (the first five exponents)
Lyapunov dimension: 2.8600

case 2

parameters: A = 16.0, B = 1.0, C = π/3, τ = 0.28, DIM = 100
The step size of the fourth-order Runge-Kutta method: τ/DIM = 0.0028
Lyapunov exponents (log with base-e): 0.4532, -0.0595, -3.3951, -5.9213, -6.5976 (the first five exponents)
Lyapunov dimension: 2.1159