ATRACTOR

This was an example that turned out to be very interesting and that it is not usually dealt with. The difference in relation to the previous one is that it is now assumed that the rigid bar is replaced by a string (a line), which does not allow the ball to move away more than L (length of the pendulum) from the support point, but allows it to move away less than L (unlike the previous example).

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For initial rest positions between -90º e 90º, the behavior is similar to the previous one, but, between 90º e 180º for example, the movement begins in free fall, subject only to the action of gravity, until the distance equals L, then giving a jerk.

It was necessary to make a hypothesis regarding the nature of this jerk, in order to program what happens next: it was assumed that the shock was totally inelastic. The consequent loss of energy can be confirmed and interesting phenomena can be discovered, for example what happens for starting amplitudes slightly above 90º: a series of alternations between short intervals of true pendulum movement and others of free fall (until the string is totally stretched out), with a slow decrease in total energy. There is also an option that allows you to control the resistance of the string: it may break if it is not very resistant…
In this case, it would not make sense to talk about the phase plane and the phase cylinder (in «free fall» the position space already has dimension 2), so the respective tables do not appear.

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