# Solve and plot Lorenz equations for two different initial conditions and two values of rho in julia

**URL:** <https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366>\
**Category:** New to Julia\
**Tags:** question, plotting\
**Created:** [March 1, 2023, 6:06am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366 "2023-03-01T06:06:02Z")\
**Posts on this page:** 8\
**Page:** 1

<div class="post-metadata">

**Author:** ![Hana](https://avatars.discourse-cdn.com/v4/letter/h/b19c9b/32.png) [@Hana](https://discourse.julialang.org/u/Hana)\
**Post date:** [March 1, 2023, 6:06am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/1 "2023-03-01T06:06:02Z")

</div>

My goal is to solve lorenz equations and plot them as it shows in the figure. I have two different initial conditions [x0, 1, 0] and x0= 0 then x0 =1\* 10^-5 the two values of rho are ρ= 14 and ρ=28. I wrote this code in julia but I’m getting errors says “Got exception outside of a @test UndefVarError: u not defined lorenz\_solver(x₀::Float64, ρ::Int64)” any idea on how to solve this issue.

![image](https://global.discourse-cdn.com/julialang/original/3X/b/5/b5c6f908e9b5cf96a93e95886aa9569612b92e24.png)

```julia
using DifferentialEquations
using Plots
using LaTeXStrings

function lorenz!(t,p,u,du)
    x, y ,z = u
    sigma, ρ, beta = p
    du[1] = sigma*(u[2]-u[1])
    du[2] = u[1]*(ρ-u[3]) - u[2]
    du[3] = u[1]*u[2] - beta*u[3]
end

function lorenz_solver(x₀, ρ)
    u[0] = [x₀, 1, 0]
    tspan = (0.0, 100.0)
    p = [10, ρ, 8/3] 
    x₀, ρ = params
    prob = ODEProblem(lorenz!, u0, tspan, params)
    sol = solve(prob) 
    return sol
    #return the ODESolution
end

function lorenz_plot()
    params = [0, 14]
    sol1 = solve(prob)

    params = [1 * 10^-5 , 14]
    sol2 = solve(prob)

    p1 = plot(t, [sol1 sol2], ylabel=L"x1, x2", title="ρ = 14")
    p2 = plot(abs(sol1 - sol2), xlabel=L"t", ylabel=L"|x1 - x2|")
    p3 = plot(sol1, vars = (1,2,3), xlabel=L"x1", ylabel=L"z1")

    params = [0, 28]
    sol3= solve(prob)
   
    params = [1 * 10^-5, 28]
    sol4 = solve(prob)

    p4 = plot(t, [sol3 sol4], title="ρ = 28")
    p5 = plot(abs(sol3 - sol4), xlabel=L"t")
    p6 = plot(sol3, vars = (1,2,3), xlabel=L"x1")
    link=:all
    plot(p1, p2, p3, p4, p5, p6, layout(3, 2))
    #return the final plot
end

export lorenz_solver, lorenz_plot

end

```

---

<div class="post-metadata">

**Author:** ![tomaklutfu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tomaklutfu/32/2411_2.png) [@tomaklutfu](https://discourse.julialang.org/u/tomaklutfu)\
**Post date:** [March 1, 2023, 7:17am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/2 "2023-03-01T07:17:52Z")

</div>

> [@Hana](#):
>
> `u[0] = [x₀, 1, 0]`

Here you might have wanted to type `u0 = [x₀, 1, 0]`.

---

<div class="post-metadata">

**Author:** ![Hana](https://avatars.discourse-cdn.com/v4/letter/h/b19c9b/32.png) [@Hana](https://discourse.julialang.org/u/Hana)\
**Post date:** [March 1, 2023, 7:23am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/3 "2023-03-01T07:23:19Z")

</div>

I just changed it and got an error says " Got exception outside of a @test  
BoundsError: attempt to access 2-element Vector{Float64} at index [3]"

---

<div class="post-metadata">

**Author:** ![tomaklutfu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tomaklutfu/32/2411_2.png) [@tomaklutfu](https://discourse.julialang.org/u/tomaklutfu)\
**Post date:** [March 1, 2023, 7:40am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/4 "2023-03-01T07:40:47Z")

</div>

> [@Hana](#):
>
> ```julia
> p = [10, ρ, 8/3] 
> x₀, ρ = params
> prob = ODEProblem(lorenz!, u0, tspan, params)
> 
> ```

You have 2 different things as parameters. 1 for the general problem you want to solve `(x₀, ρ)` and 1 for the ODE problem of Lorenz equation. For the ODE problem, you should put `p` as an argument not `params`.

---

<div class="post-metadata">

**Author:** ![Hana](https://avatars.discourse-cdn.com/v4/letter/h/b19c9b/32.png) [@Hana](https://discourse.julialang.org/u/Hana)\
**Post date:** [March 1, 2023, 7:47am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/5 "2023-03-01T07:47:05Z")

</div>

@tomaklutfu  
I’m not sure if I’m getting since I’m new with Julia.  
I added p as an argument here:

function lorenz!(t,p,u,du)

then I defined sigma and beta to solve the problem while keeping ρ undefined until the plots section since it’s going to take 2 different values

---

<div class="post-metadata">

**Author:** ![tomaklutfu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tomaklutfu/32/2411_2.png) [@tomaklutfu](https://discourse.julialang.org/u/tomaklutfu)\
**Post date:** [March 1, 2023, 7:57am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/6 "2023-03-01T07:57:56Z")

</div>

> [@Hana](#):
>
> ```julia
> p = [10, ρ, 8/3] 
> x₀, ρ = params
> prob = ODEProblem(lorenz!, u0, tspan, params)
> sol = solve(prob) 
> 
> ```

The problem is in the construction of the `ODEProblem`, especially in the above quote. Confusing yourself with the parameter of the general question in your hand, you pass a 2-valued (`x₀` and `ρ`) variable named `params` as an argument to it. In fact, the parameter that should be passed is 3-valued `p` ([10, ρ, 8/3]) instead.

---

<div class="post-metadata">

**Author:** ![RayleighLord](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rayleighlord/32/33592_2.png) [@RayleighLord](https://discourse.julialang.org/u/RayleighLord)\
**Post date:** [March 1, 2023, 8:27am UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/7 "2023-03-01T08:27:02Z")

</div>

You should read [this section](https://docs.sciml.ai/DiffEqDocs/stable/getting_started/#Example-2:-Solving-Systems-of-Equations) of the tutorials carefully. The order of the arguments that you are using when defining the `lorenz!` function is not what the `DifferentialEquations` package expects.

If you check out the tutorial, you will see

```julia
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

```

That is the order of arguments you shoud use. Now, if you want to start simulating the system for different parameters, you can pass them via the `p` vector, but always defining the function as `lorenz!(du,u,p,t)` and not defining the arguments in a different order.

In fact, a little below on the [tutorial](https://docs.sciml.ai/DiffEqDocs/stable/getting_started/#Defining-Parameterized-Functions) you actually have what you want, the lorentz system parametrized.

---

<div class="post-metadata">

**Author:** ![Sergey\_Novak](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sergey_novak/32/37716_2.png) [@Sergey\_Novak](https://discourse.julialang.org/u/Sergey_Novak)\
**Post date:** [March 2, 2023, 6:33pm UTC](https://discourse.julialang.org/t/solve-and-plot-lorenz-equations-for-two-different-initial-conditions-and-two-values-of-rho-in-julia/95366/8 "2023-03-02T18:33:47Z")

</div>

I want to add to the last answer that it is necessary to keep track of what type of variables you are passing. Everything should be, for example, Float32. This can greatly affect performance