Add DDE support in System

[YingboMa]

@parameters p0=0.2 p1=0.2 q0=0.3 q1=0.3 v0=1 v1=1 d0=5 d1=1 d2=1 beta0=1 beta1=1
@variables t x₀(t) x₁(t) x₂(..)
tau = 1
D = Differential(t)
eqs = [D(x₀) ~ (v0 / (1 + beta0 * (x₂(t - tau)^2))) * (p0 - q0) * x₀ - d0 * x₀
    D(x₁) ~ (v0 / (1 + beta0 * (x₂(t - tau)^2))) * (1 - p0 + q0) * x₀ +
            (v1 / (1 + beta1 * (x₂(t - tau)^2))) * (p1 - q1) * x₁ - d1 * x₁
    D(x₂(t)) ~ (v1 / (1 + beta1 * (x₂(t - tau)^2))) * (1 - p1 + q1) * x₁ - d2 * x₂(t)]
@named sys = System(eqs)
prob = DDEProblem(sys,
    [x₀ => 1.0, x₁ => 1.0, x₂(t) => 1.0],
    tspan,
    constant_lags = [tau])
sol_mtk = solve(prob, alg, reltol = 1e-7, abstol = 1e-10)
@test sol_mtk.u[end] ≈ sol.u[end]
prob2 = DDEProblem(sys,
    [x₀ => 1.0 - t * q1, x₁ => 1.0 - t * q1, x₂(t) => 1.0 - t * q1],
    tspan,
    constant_lags = [tau])
sol2_mtk = solve(prob2, alg, reltol = 1e-7, abstol = 1e-10)

now works. See test/dde.jl for more details.

[ChrisRackauckas]

bold assumption. yeah we'll need to have some way to give a history function.

[YingboMa]

Since it's a positional argument before something with a default, it must also have a default.

[ChrisRackauckas]

What is your planned representation for passing symbolic history functions? Just if the default value is a symbolic expression?

[YingboMa]

Yeah, since h(p, 0) is just u0, right?

[ChrisRackauckas]

Yes, so okay just wanted to double check.