The pendulum in the forced-absorbed oscillations








Equations

Now, we consider that the friction coefficient is not a constant. It changes in respect of the position. For the pendulum, it means that it is accelerated  for small angles and slowed down for large angles. The limit angle where there is no forcing or friction depend on the set of parameters: mu and k.

The equation is known as the Van der Pol attractor:

It gives the following system:



Results

It leads to attractors that are considered as "strange". The attractors correspond to a type of oscillation. Indeed, the pendulm can't stop in a zero position because of the excitation, but the movement cannot diverge because of the friction present for high angles.

    

From two different initial conditions, the result is the same type of oscillation. If we look closely, the attractor is not circular or ellipsoidal and the evolution of the angle in time does not follow a sinusoidal.

     

Here it is almost the same but for a different set of parameters k and mu.