Part G: Numerical Integration and Visualization

Exercises

The dynamics homework this week asks you to explore various two-dimensional maps and ODEs and the behaviors they produce. Use the simulation and display programs introduced in this part of the tutorial for these problems by modifying them appropriately. For the ODE problems, use the Runge-Kutta integrator that you are asked to write below.
  1. Write a 3D fourth-order Runge-Kutta integrator. Make it a function RK3DIntegrator() that you can call, passing in the ODE functions to be integrated and other simulation parameters, such as the number of time steps to integrate over and the size of the time step.
    Hint: Start with the 2D Runge-Kutta integrator discussed in class (and on the class website) and extend it to three variables. Recall the Lecture Notes on numerical integration, which describe the 3D Runge-Kutta method.
  2. Modify the MayaVi program LorenzODE.py so that its Euler integrator is a function Euler3DIntegrator() that is called with the ODEs and simulation parameters, as above.
    Hint: If you are having trouble running MayaVi, then, as an alternative, modify the matplotlib version of the Lorenz program LorenzODE.2d.py.
  3. Integrate the 3D fourth-order Runge-Kutta integrator RK3DIntegrator() into the above program in such a way that you can easily switch between the Euler and Runge-Kutta integrators. That is, have the simulator program call one or the other integrator using a generic ODEIntegrator() function that is set appropriately to one or the other integrator function.
  4. Graphically compare the accuracy of the Euler and Runge-Kutta methods for the Lorenz simulation. That is, compare them at the same integration time-step size. How do the solutions differ? How would you convince yourself (or others) that one is more accurate than another?
    Hint: Recall the similar comparison that we did in the programming lab between using the Euler, improved Euler, and Runge-Kutta methods for 1D ODEs.
  5. Two-dimensional map orbit tool: Make a collection of two-dimensional map functions (investigated last week) that can be used by a general 2D map orbit display program. Do this by making a library that includes all of the two-dimensional map functions. Store this in a separate *.py file. Have the program load this library (like any module). Modify the program so that where it called the Henon map function, it instead calls a generic TwoDMap() function with the appropriate number of parameters. Also, include the various defaults (parameters, initial conditions, and plot labels).
    Hint: Recall that we did an analogous task for 1D map bifurcation diagram tool in the previous programming lab exercises. See the steps outlined there for details of what to include.
  6. Do this same generalization with the ODE programs, so that you can easily switch between different sets of ODEs, including Lorenz and Rossler ODEs.
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