Homework for Garcia Ch. 3, orbital motion
Play with orbit.m! Get familiar with the rk4 and rka function calls inside orbit.m, and with
the derivatives computed by gravrk.m. Look at rk4.m and rka.m themselves to peak under the hood.
Interesting test parameters might include:
- Editing the eccentricity to be more circular (smaller)
- Initial distances of 1, 2 AU or similar
- Initial velocities in multiples of pi (1, 2, etc)
- Timesteps of 0.02, 0.01 yr or smaller
- 100-200 steps
If your comet flies away, try decreasing the time step or going adaptive!
Upload an image of the coolest plot you make.
Modify the program projectile.m and the associated function derivsProj.m so that it includes the option to use the Runge-Kutta (rk4.m) and adaptive Runge-Kutta (rka.m) solvers. Your program is working if it produces the following outputs for the three methods (hit "Run" multiple times and add a legend in the command window).
What conclusions can you make about the different solvers based on your results? Use the disp() function to have your program print your answer in the command window.
(From Garcia, #25 page 90) One characteristic of chaotic dynamics is sensitivity to initial conditions.
First, complete the lorzrk.m function to compute the derivatives needed for the Lorenz Attractor.
Then edit lorenz.m to include the non-adaptive version of the Runge-Kuatta method. Update the code so that
it simultaneously computes the trajectories for two different sets of initial conditions.
- Use initial conditions that are very close together (ex: [1 1 20.000] and [1 1 20.001])
- Carry out the computation for 3000 steps with a time step of 0.005 seconds
- Use parameter values of sigma = 10, b = 8/3, and r = 28
- Plot the distance between these trajectories as a function of time
How does the distance vary with time? How does this relate to chaotic dynamics? Do your conclusions change at all when using the adaptive Runge-Kutta method? Use the disp function to have your program print your answer in the command window.