Why no chaos for continuous systems with two degrees of freedom
Let's explain better why a continuous planar orbit cannot be chaotic.
A discrete planar orbit will see the point representing the system
jumping in the plane. The orbit can be periodic with the same points
visited again and again. Or it can be chaotic:this means that the
same point is never visited again.In practice since the computer plane
is finite and not infinite,the orbit must eventually return on the same point
after a very,very long cycle. Eventually,if the orbit visits all the
computer points the cycle will be equal to the number of possible points and
this is connected to the precision of the numbers used.If the orbit is
continuous,also in the computer discrete plane the next point is always one
of the 8 nearest points . If you draw a line on a plane like a paper sheet,
it will first or later cross itself! This means that it is a periodic orbit
since from the crossing on, it will follow the previous orbit.
For this
reason you cannot have a continuous chaotic orbit in the plane.
Instead in space you can draw continuous lines that can continue forever (at
last in the ideal mathematical space) without ever crossing themselves:this
is for example what happens in the Lorentz attractor.
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