The driven pendulum
Once an excitation force is applied on the pendulum, the chaos can born. Indeed, even if there is a non-linear term in the equation, it has been shown that there is no chaotic behavior for a simple or damped pendulum which is not driven.The pulsation wf
In this part, we can see that as soon as a third variable is added, the problem becomes still more complex. We are going to see the phase portraits of the pendulum in function of the pulsation and the amplitude of the excitation force.
On the charts below, we can see three parts in function of wf.
The first, at the , seems to be chaotic, but it is not a real chaos. Indeed, we can notice two strange attractors. The is describing one turn before going back and doing another round in the opposite direction. The has got the same behavior but adds two "back'n run", that is to say that the pendulum is going a few back before going on is loop. Moreover, this second attractor seems to lead to chaos as , but it is not the case as we can see on the 3D phase graph.
The second, in the middle, is governed by an strange attractor.
The third, at the top, is significant of a excitation pulsation. The pendulum is accelerated too often.
The amplitude b
For the amplitude, it seems to be two parts, one is "regular" whereas the second would be chaotic. Nevertheless by watching at the results, it appears that the is well governed by a strange attractor, but the second is more difficult to treat. Indeed, there are several strange attractors and also behaviors. The strange attractors noticed are for the following amplitudes : , , . With the charts below, it is possible to see the different phase graphs by moving the mouse on the selected amplitude.
Driving puts energy into the system which is dissipated by the viscous damping term. Thus, energy is pumped through the system. On this chart, we can see there is a balance between the work done at the pendulum and the dissipated energy, until the amplitude of the excitation force becomes too important.. And so, leads to chaos.
To conclude, hereunder is a last example of chaotic behavior :