As the graph shows the initial position of the two

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As the graph shows, the initial position of the two “Earths” are exactly the same, but they have a slight difference in velocity, say, one has a velocity of 0.001 and the other has a velocity of 0.0010001. To the above is the graph of trajectories of the two conditions. From the animation, we can find that at the beginning, the trajectories matches fairly the same, this is what we expect for a non-chaotic system; but suddenly, in a later phase, the trajectories diverge radically. T his radical divergence arising from minuscule differences (which is 0.1% in this case) in the initial conditions is known as Sensitive Dependence on Initial Conditions, which is one of the most primary characteristic of all chaotic systems. Also, this phenomenon is known as the “Butterfly Effect”. With the examples shown above, the Three Body Gravitational Problem shows a lot of characteristics of a chaotic system. Like Sensitive Dependence on Initial Conditions and Non- repeatable Trajectory: slight difference in initial point will lead to great difference in later trajectory, and the trajectory never repeats. These evidences clearly show how and why Three Body Gravitational Problem behaves as chaotic system. Words count: 561 References: Wikipedia: Lecture notes
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