In 1963 Edward Lorenz derived a simple set of equations describing convection in the atmosphere:

dx/dt = σ(y-x),

dy/dt = x(Ρ-z)-y

dz/dt = xy - βz

Even though these equations are simple and deterministic, long-term behavior of solutions for some particular values of parameters (e.g. σ = 10, ρ = 28, β = 8/3) could be highly unpredictable. Small variations of initial conditions could result in drastic difference of the corresponding solutions of the system. The latter phenomenon is known as the butterfly effect: small perturbations of the atmosphere caused by the butterfly wings at one location on Earth can result (according to the model) in substantial changes in the atmosphere at another location.

Consider (1) and let the initial conditions and the interval of interest be as follows

x(0) = 0, y(0) =1, z(0) = 0, t  ∈ [0,50].

Problem 1:

Please implement Euler integration scheme for (1) with integration step Δ = 0.0001.

Problem 2:

Please implement improved Euler integration scheme for (1) with Δ = 0.0001.

Problem 3:

Please implement Runge-Kutta integration scheme for (1) with Δ = 0.0001. Change x(0) = 0 to x(0) = 0.0001.

Observe the butterflieffect.

Problem 4:

Please quantify how different are these numerical solutions from each other (take Runge-Kutta scheme with x(0) = 0 as a reference and plot their differences). Investigate what happens with these differences when the value of Δ is made 5 times smaller/larger.