In 1962 Ed Lorenz was simulating simple models of atmospheric convection. He discovered that such models could evolve such that states that start close together can, after some time, be well separated. Lorenz termed this the "butterfly" effect and such systems are now termed chaotic. The atmospheric flow shows chaotic behaviour. The Lorenz Model is a set of coupled 1s Order ODEs. dx dt dy dt where (Prandtl Number, p and (Rayleigh Number) are parameters of this model with "default" values of 10, 8/3 &28 respectively One way to implement this model in Python is to code a function LorenzDerivatives (Xut) which takes: Xi a list of values corresponding to the initial x, y and z.' whilst the time is not required to calculate the derivative- And then returns the derivatives (a vector of values corresponding to x', y' and z') for the Lorenz Model. . You then give this function to ode int (Lorenz Derivatives. Xrt) which solves it giving you solutions at the times you request. How does the rate at which initially close trajectories diverge depend on the Rayleigh umber when it is near 28? By divergence I mean the rate at which the distance grows - ou should see that that the distance initially grows exponentially before entering a chaotic ehaviour. If you see exponential growth in the distance, you can identify the exponent to uantify the rate of divergence. irst, you will need to create a variable which quantifies the distance between 2 trajectories nd use this to plot how the distance changes with time. You should explore this with the orenz model for a variety of initial conditions and parameters to get some understanding to ts behaviour.. Asignature of a chaotic system is that the evolution of a system is very sensitive to the nitial conditions so that uncertainty in the initial state of a system, such as the weather neans that it rapidly becomes difficult to forecast. a. Choose a way to investigate this problem; write a short description saying how you will nalyse it. b. Implement your design in Python/Ssipy/Numpy c. Investigate and come to some conclusions. In 1962 Ed Lorenz was simulating simple models of atmospheric convection. He discovered that such models could evolve such that states that start close together can, after some time, be well separated. Lorenz termed this the "butterfly" effect and such systems are now termed chaotic. The atmospheric flow shows chaotic behaviour. The Lorenz Model is a set of coupled 1s Order ODEs. dx dt dy dt where (Prandtl Number, p and (Rayleigh Number) are parameters of this model with "default" values of 10, 8/3 &28 respectively One way to implement this model in Python is to code a function LorenzDerivatives (Xut) which takes: Xi a list of values corresponding to the initial x, y and z.' whilst the time is not required to calculate the derivative- And then returns the derivatives (a vector of values corresponding to x', y' and z') for the Lorenz Model. . You then give this function to ode int (Lorenz Derivatives. Xrt) which solves it giving you solutions at the times you request. How does the rate at which initially close trajectories diverge depend on the Rayleigh umber when it is near 28? By divergence I mean the rate at which the distance grows - ou should see that that the distance initially grows exponentially before entering a chaotic ehaviour. If you see exponential growth in the distance, you can identify the exponent to uantify the rate of divergence. irst, you will need to create a variable which quantifies the distance between 2 trajectories nd use this to plot how the distance changes with time. You should explore this with the orenz model for a variety of initial conditions and parameters to get some understanding to ts behaviour.. Asignature of a chaotic system is that the evolution of a system is very sensitive to the nitial conditions so that uncertainty in the initial state of a system, such as the weather neans that it rapidly becomes difficult to forecast. a. Choose a way to investigate this problem; write a short description saying how you will nalyse it. b. Implement your design in Python/Ssipy/Numpy c. Investigate and come to some conclusions