hw7 - Sigma and b are fluid properties of the air and r is...

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Physics 503: Scientific Computing Homework #7 Topic: Solutions of ODEs: the Lorenz climate model exhibiting Chaos! Due: Thurs. 4/1 by the beginning of class - DON’T forget to include output and plot printouts! (please email your code to [email protected] ) Assignment 1. Use the Euler method to solve the system of ODEs dx dt = σ x + y dy dt = xz + rx y dz dt = xy bz for = 10.0 , r=29, and b = 8.0/3.0 and starting points: x 0 = 1.0, y 0 = 1.0, z 0 = 20.1 . Make plots of x,y, and z as a function of time. If you are feeling ambitious, create a 3D phase space plot where x,y,z are the variables. (You can use the 3D plotting capabilities of pylab like this: from mpl_toolkits.mplot3d import Axes3D fig3d=figure(3) ax=Axes3D(fig3d) ax.plot3D(x,y,z) 2. Determine the fixed points for this system of equations. (Pencil and paper or Mathematica) A little background This system was originally developed by Lorenz to model buoyant convections in the atmosphere.
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Unformatted text preview: Sigma and b are fluid properties of the air and r is the applied temperature gradient. The variables are: x – rate of convective overturning, y & z – horizontal and vertical temperature gradients. He found very interesting mathematical behavior which quickly led to a new field of mathematics called Chaos Theory. One of the hallmarks of chaos is that very small changes in the initial conditions quickly lead to dramatically different time evolution of the variables. The fixed points in this model are called ‘strange attractors’ because the attract the variables toward them, but the system never settles down into an equilibrium state. You should be able to quickly take code we looked at in class and rework it to solve the Lorenz equations....
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