delay
A delay differential equation is an ODE which allows the use of previous values. In this case, the function needs to be a JIT compiled Julia function. It looks just like the ODE, except in this case there is a function h(p,t) which allows you to interpolate and grab previous values.
We must provide a history function h(p,t) that gives values for u before t0. Here we assume that the solution was constant before the initial time point. Additionally, we pass constant_lags = c(20.0) to tell the solver that only constant-time lags were used and what the lag length was. This helps improve the solver accuracy by accurately stepping at the points of discontinuity. Together this is:
f = JuliaCall::julia_eval("function f(du, u, h, p, t)
du[1] = 1.1/(1 + sqrt(10)*(h(p, t-20)[1])^(5/4)) - 10*u[1]/(1 + 40*u[2])
du[2] = 100*u[1]/(1 + 40*u[2]) - 2.43*u[2]
end")
u0 = c(1.05767027/3, 1.030713491/3)
h <- function (p,t){
c(1.05767027/3, 1.030713491/3)
}
tspan = list(0.0, 100.0)
constant_lags = c(20.0)
sol = diffeqr::dde.solve('f',u0,h,tspan,constant_lags=constant_lags)
udf = as.data.frame(sol$u)
plotly::plot_ly(udf, x = sol$t, y = ~V1, type = 'scatter', mode = 'lines') %>% plotly::add_trace(y = ~V2)delay
Notice that the solver accurately is able to simulate the kink (discontinuity) at t=20 due to the discontinuity of the derivative at the initial time point! This is why declaring discontinuities can enhance the solver accuracy.