These figures show a primary numerical calculation of almost invariant sets.

The three three-dimensional shapes are the Lorenz attractor viewed from three different angles.

Because our constant is varying from impact to impact, the invariant measure is different to before.

This means that the frequency of table phase/exit velocity combinations has changed in a precise way.

Consider a ball bouncing on a sinusoidally forced moving table.

Let \( s_t \) represent the phase of the table at impact at time \( t \), and \( v_t \) represent the exit velocity of the ball immediately after impact with the table.Invariant sets are very important, as each one is a building block of the system.In chaotic dynamical systems, there are no invariant sets; there is no way to break the system up into simpler pieces.The best that one can do is to look for almost-invariant sets.These sets break the system up into pieces which are individually highly chaotic, but which have only weak chaotic connections between themselves.If we draw a curve \( (x(t), y(t), z(t))\) in three-dimensional space, we will trace out a strange attractor known as the Lorenz attractor.An invariant set is a piece of an attractor that does not move in time.This distribution is also invariant under the evolution of the dynamical system.Being invariant means that this distribution looks the same, no matter when you observe it.\end The constant \( a \) is a factor between 0 and 1 describing how “bouncy” the ball is, and the constant \( g \) is a positive number describing the force applied by the table.The bouncing ball evolution defined by the above formula is an example of a dynamical system.

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