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")
h <- JuliaCall::julia_eval("function h(p, t)
[1.05767027/3, 1.030713491/3]
end")
u0 <- c(1.05767027/3, 1.030713491/3)
tspan <- c(0.0, 100.0)
constant_lags <- c(20.0)
JuliaCall::julia_assign("u0", u0)
JuliaCall::julia_assign("tspan", tspan)
JuliaCall::julia_assign("constant_lags", tspan)
prob <- JuliaCall::julia_eval("DDEProblem(f, u0, h, tspan, constant_lags = constant_lags)")
sol <- de$solve(prob,de$MethodOfSteps(de$Tsit5()))
udf <- as.data.frame(t(sapply(sol$u,identity)))
plotly::plot_ly(udf, x = sol$t, y = ~V1, type = 'scatter', mode = 'lines') %>% plotly::add_trace(y = ~V2)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.