The pendulum is an important system because throughout history it has been the emblem of classical mechanics and the epitome of clockwork regularity.  It was used by Galileo in his church lamp experiment to describe friction and force and Huygens in timekeeping.  Foucault used it to demonstrate the rotation of the earth.  For a system that appears to be so simple, in reality is much more comlex.

            The following simulation of the pendulum demonstrates the period-doubling route to chaos with the use of Time Series, Phase Diagrams, Poincaré Sections, Power Spectrums, and Bifurcation Diagrams.  Through these methods, the dynamics of the system can be understood. 

 

Reinitialize

Equation of Motion:

            According to Newton’s second law, the equation of motion for the damped, driven pendulum of mass m and length l can be written as the following:

 

(1)1

The various terms on the left represent acceleration, damping, and gravitation.  The term on the right is the driving force.  The angular driving force ωD does not have to be equal to the natural frequency of the system. 

By changing the unit of time to the following:

 

(2)2

and substituting it into equation 1, the equation of motion becomes:

 

(3)2

Setting mlτ-2 = mg, we can then define τ as the following:

 

(4)2

and when we divide through by mgl, we get the the following dimensionless equation of motion:

 

(5)1

q is the dampening parameter and g is the forcing amplitude.  Depending on the values of these parameters, the system can exhibit dynamics ranging from periodic to chaotic.  Table 1 depicts the various characterisitcs for the system given g while ωD = 2/3 and q = 2.

 

Range

 

Type of Behavior

 

 

            g < 1.085

 

periodic

 

 

1.085 < g < 1.11

 

chaotic

 

 

1.11   < g < 1.14

 

periodic

 

 

1.14   < g < 1.22

 

chaotic

 

 

             g ~ 1.22

 

periodic

 

 

1.22   < g < 1.28

 

chaotic

 

 

1.28   < g < 1.475

 

periodic

 

 

1.475 < g < 1.485

 

chaotic

 

 

1.485 < g < 1.493

 

periodic

 

 

1.493 < g < 1.495

 

chaotic

 

 

1.495 < g < 1.497

 

periodic

 

 

1.497 < g

 

chaotic

 

Table 11:  Type of Behavior exhibited for values of g ranging from 0.9 to 1.50.  (Table can be found in Baker and Gollub)

            The rate equations used to step the system are derived from the dimensionless equation of motion (eq. 5) and integrated using the Runge-Kutta algorithm.

 

(6a)1

 

(6b)1

 

(6c)1

φ is the phase of the driving term.  The three dimensions for this system then become ω, θ, and φ.  To simplify the results of the system, θ has been restricted to reside within –π and +π.  φ has been restricted to reside within 0 and 2π.  These changes allow for better viewing of the Poincaré and Phase Diagrams.

Control:

           There are six changeable parameters available in this system.  These parameters are Omega, Theta, Phi, q, g, and the Poincare Section.  Each of these variables represents a different feature of the applet and can be defined by the user.

           The parameters Omega, Theta, and Phi represent the state of the current pendulum.  When the panel is in edit mode, the user may change the initial conditions of the pendulum.  If the pendulum is stopped or stepped, the input window will display the current state of the pendulum.  The two parameters below the pendulum state are the dampening parameter (q) and the forcing amplitude (g).  If the control panel is in edit mode, the user can change the parameters for the system.  The last parameter in the input window is the “Poincare Section.”  This parameter corresponds to the angle of the Poincaré Section observed.  Since the Poincaré Section is a slice of the Phase Space at a given angle of the driving force, this allows the user to view different sections of the phase space.

            The button panel contains the methods of the control.  This is where the user can actually start and stop the animation of the pendulum.  The custom button that has been added to the control is the “New Pendulum” button.  This button can only be accessed if the control is in edit mode and can create additional pendulums in the system.  The new pendulum is created at the position defined by the state in the input window.  

            The message window displays the unchangeable parameters of the system.  These two parameters are the time step and the driving frequency (ωD).  The time step is set to 3π/100 and ωD is set to 2/3.

References:

1Gregory L. Baker, Jerry P. Gollub.  Chaotic Dynamics: an introduction.  Cambridge University Press, NY; 1996. 

2Brian Davies.  Exploring Chaos: Theory and Experiment.  Perseus Books, MA; 1999.