Data:
Analysis
In Tables 1A and 1B, I recorded the coordinate plane position of the first mass (the red bob in the Figures 1-4). In Tables 2A and 2B, I recorded the positions of the second mass (the blue bob in Figures 1-4). I used these tables to create Figures 5 and 6, which show how the differences in coordinate points increase between the control (a double pendulum with a starting angle of π/2) and double pendulums with the other starting angles of 2.0001, 2.001, and 2.01. As you can see, all of these differences increase. This increase shows high sensitivity to initial starting conditions because unlike other systems in which a change will remain constant, a small difference here in initial conditions will continue to grow and eventually have a large impact on the path of the pendulum. For example, the pendulum with the initial starting angle of π/2.0001 took what appears to be the same path as the control for the first 10 seconds, after which the difference started to climb steeply and at 30 seconds there was a 1.25 and 0.31 difference in the x and y coordinates, respectively. Another observation is that the pendulums with a larger difference in the initial starting angle usually increased quicker and more significantly, but that was not true for all of the cases (ex. in Figure 6A π/2.0001 grew the most). This could be due to the dense periodic orbit and topological transitivity of the double pendulum, which allows the states to move arbitrarily close to or far away from a point.
Figures 1-4 show the paths of pendulums with initial starting angles of π/2, π/2.0001, π/2.001, and π/2.01 at 30, 25, 20, 15, 10, 5, and 0.01 seconds. Most of these paths start diverging more visibly after 10 seconds, and as shown by the previous tables and graphs, the paths become more and more different.
Figures 7A-E show the difference between the angles of mass one and mass two of two double pendulums with different initial velocities of mass two over time. The green is the first mass and the blue is the second. The climb is not very steady, but it does gradually increase as time goes on. Again, the unsteady climb is most likely unsteady because of the topologically transitivity and dense periodic orbits of the double pendulum.
All of these tables and graphs showed an increase in difference over time, proving that a double pendulum is sensitive to initial starting conditions.
Figures 1-4 show the paths of pendulums with initial starting angles of π/2, π/2.0001, π/2.001, and π/2.01 at 30, 25, 20, 15, 10, 5, and 0.01 seconds. Most of these paths start diverging more visibly after 10 seconds, and as shown by the previous tables and graphs, the paths become more and more different.
Figures 7A-E show the difference between the angles of mass one and mass two of two double pendulums with different initial velocities of mass two over time. The green is the first mass and the blue is the second. The climb is not very steady, but it does gradually increase as time goes on. Again, the unsteady climb is most likely unsteady because of the topologically transitivity and dense periodic orbits of the double pendulum.
All of these tables and graphs showed an increase in difference over time, proving that a double pendulum is sensitive to initial starting conditions.