So I have been recently reading about chaotic motion in the solar system. Some research articles particularly emphasize on sensitivity to initial conditions as a prime governing factor behind the rise of a chaotic situation. But what does that physically mean for let's say our solar system? I may be asking a really question, but one thing that I am just not getting is how exactly can I make sense of sensitivity to initial conditions for our planetary motion?
An easier to understand example of chaos is a frictionless billiard table. Suppose you have several perfectly spherical billiard balls in a perfectly straight line. Suppose you shoot the end ball perfectly straight toward the next ball. All the balls will bounce off each other. The end ball bounces off the table and returns to the next ball. All the balls will stay on the line forever.
Suppose your aim is slightly off. The first ball hits the second slightly off center on the left. The second ball is deflected slightly off the line to the right. In this way, each ball is deflected. The longer you watch, the farther each ball travels from the line. When struck again, the misalignment will be larger. Soon this will look nothing like the first solution.
The reason for the sensitivity to initial conditions is the curvature of the balls. A larger misalignment means a larger deflection angle. If the imperfection was a slight misalignment of one edge of the table, balls would be deflected the wrong way but the from between a perfect table would not grow as quickly.
For the solar system, several planets in nearly circular orbits isn't obviously chaotic. Replace them with many planets in random initial directions. Every so often two planets will pass close to each other and deflect each other strongly. Just how strongly depends on exactly how close they pass and their relative velocities. Small changes in these parameters make a large different in deflection.
If you imagine a single sun with a single satellite, the orbital dynamics can be described quite simply. That's because the sun is only exerting a gravitational field on the satellite and the satellite is only exerting it's gravity on the sun. So they'll orbit about their gravitational center. When you introduce a second satellite, each body exerts gravity onto each other body. This makes each satellite either slow down or speed up about the sun. This also means that each satellite will also either get closer to the central mass or further away over time [like if the two satellites are both on one side of the sun or on opposite sides]. It then follows that the position and movement of each body effects the position and movement of each other body. Because of the back and forth requirement for calculating how everything moves, you have to have a starting point from which you calculate the motion of things, which then allows you to calculate the position of things, which allows you to calculate the movement of things etc. This also means that, to a certain degree, the chaotic nature of the system is not dependent on the number of bodies. That is to say, 4 orbiting bodies isn't inherently more chaotic than 5 orbiting bodies. There's more to calculate, but it's not more chaotic.