I define the Lorenz system twice, once the standard way and once with an extra equation that depends only on u_1.

```
using DifferentialEquations
using Makie, CairoMakie
function lorenz!(du, u, p, t)
du[1] = 10.0 * (u[2] - u[1])
du[2] = u[1] * (28.0 - u[3]) - u[2]
du[3] = u[1] * u[2] - (8 / 3) * u[3]
end
u0 = [1.0; 0.0; 0.0]
tspan = (0.0, 100.0)
prob = ODEProblem(lorenz!, u0, tspan)
sol = solve(prob, Tsit5(), abstol = 1e-12, reltol = 1e-12, dt = 1e-6)
# sol = solve(prob, Tsit5(), adaptive = false, dt = 0.01)
function lorenz2!(du, u, p, t)
du[1] = 10.0 * (u[2] - u[1])
du[2] = u[1] * (28.0 - u[3]) - u[2]
du[3] = u[1] * u[2] - (8 / 3) * u[3]
du[4] = u[1]
end
u02 = [1.0; 0.0; 0.0; 0.0]
prob2 = ODEProblem(lorenz2!, u02, tspan)
sol2 = solve(prob2, Tsit5(), abstol = 1e-12, reltol = 1e-12, dt = 1e-6)
# sol2 = solve(prob2, Tsit5(), adaptive = false, dt = 0.01)
fig = Figure()
ax = Axis(fig[1, 1])
lines!(ax, sol.t, [sol.u[i][1] for i in 1:length(sol.u)])
lines!(ax, sol2.t, [sol2.u[i][1] for i in 1:length(sol2.u)])
fig
```

It gives two radically different solutions. The question is then: can I trust the result of `lorenz2`

, and if not, how could I? This seems to have something to do with the adaptive algorithms, if you try the non-adaptive version (commented lines) you get this:

The two solutions totally overlap, which is what I naively would have expected from the first case. Of course, u'_4 as defined might be changing much faster than any of the other components, so it’s understandable that the adaptive algo might have to compensate for that, but I still don’t understand why u_1 itself wouldn’t be the same. I would also note that the same behavior is observed even if `du[4] = 0.0`

.