The dynamics of a pair of coupled pendula

Most of the information on this page is contained in the paper

M.E. Henderson, M. Levi and F. Odeh.

The Geometry and Computation of the Dynamics of Coupled Pendula International Journal of Bifurcation and Chaos, vol. 1, no. 1, (1991), pp 27-50.

Some of the results here were not included for the sake of brevity, and others are new since then.

The equations of motion of this system of coupled pendula are

which has the mechanical analog shown below
107K
The two pendula are forced with idealized "slipping" drive belts, which apply a constant torque, independant of the motion of the pendula. The coupling is through a linear torsional spring, and there is a damping on each pendulum proportional to the angular velocity of the pendulum.

There are four parameters: the two torques, the spring constant k, and the damping gamma. For most of the studies which follow, the spring constant and damping are fixed. Aronson, Doedel, and Othmer have done work on the region of small damping.

The motion of a single pendula (or two when the spring constant is zero), is well understood. For this case there are two parameters: the damping and a single torque. The figure below shows schematically what the behavior of the single pendulum is for the entire parameter space.

Steady States

Steady states, or fixed points, are determined by the equations

The steady states do not depend on the damping, so if the spring constant is fixed, the steady states lie on a surface. A peojection of the surface is shown below for several values of k.

The separation of the pendula as a function of the two torques

Grey is stable,
Yellow is unstable,
Red satisfies conditions for Shilnikov bifurcation.

k=.1 69K, k=.3 69K, k=.6 67K
Note: For fixed k, if is a steady state, so are

This accounts for the "staircase"-like appearance of the surfaces.

Figure 2.2 Section of the surface of steady states for spring constant k=.1, Difference of two torques Ic=.5*k*\pi

Singular Steady States

Singular Steady states, near which the number of steady states may change, satisfy

In this case the singularities are all saddle-node bifurctions, which are folds on the surface of steady states. The singular steady states are a two manifold in the five dimensional space (phi1, phi2, I1,I2,k), so can be shown, for example, as a surface in (I1,I2,k) space.

k as a function of the two torques

106K 133K 82K

Figure 2.4 The bifurcation set of the surface of steady states at spring constant k=.1 as a function of the two torques. Yellow lines are Ic=k\pi and -k\pi, dashed blue line is Ic=0, dashed black lines are I=0, and I=1

Figure 7.1 Four solutions on the curve for k=.1, damping=.5, Ic=.1k\pi. The trace of the two angles are plotted for the trajectory.

Periodic Motions

Periodic Motions for these coupled pendula satisfy boundary conditions

When the beginning and ending angles differ, these are called second kind, or "running" periodic motions. It can be shown that the only solutions of this type must be periodic in the difference of the two angles, that is, the average position of the pendula may "run", but the separation may not. The periodic motions may therefore be classified by the increase in the average position of the pendula over a single period.

Figure 5.1
A section of the manifold of periodic motions for k=.1, damping=.5, Ic=.1k\pi. The curve is period as a function of the average torque. Motions on the blue curve tumbles once per period, on the red curve they tumble twice 2. The half period is plotted for the red curve. The curves meet at a period doubling bifurcation. The nearly homoclinic periodic motions are show.

Figure 5.2
A section of the manifold of periodic motions for k=.1, damping=.5, Ic=.55k\pi. The curve is period as a function of the average torque. Motions on the blue curve tumbles once per period, on the red curve they tumble twice 2. The half period is plotted for the red curve. The curves meet at a period doubling bifurcation. The nearly homoclinic periodic motions are show.

Figure 5.3
A section of the manifold of periodic motions for k=.1, damping=.5, Ic=.675k\pi. The curve is period as a function of the average torque. Motions on the blue curve tumbles once per period, on the red curve they tumble twice 2. The half period is plotted for the red curve. The curves meet at a period doubling bifurcation. The nearly homoclinic periodic motions are show.

Figure 5.4
A section of the manifold of periodic motions for k=.1, damping=.5, Ic=.975k\pi. The curve is period as a function of the average torque. Motions on the blue curve tumbles once per period, on the red curve they tumble twice 2. The half period is plotted for the red curve. The curves meet at a period doubling bifurcation. The nearly homoclinic periodic motions are show.

Here are bifurcation diagrams (slices of the manifold of periodic motions), for fixed spring constant k and damping, at various values for the difference of the two torques (numbers are Ic divided by k\pi).
0. 0. 0. 0.05 0.05 0.10 0.10 0.10 0.15 0.15 0.20 0.20 0.25 0.25 0.30 0.35 0.40 0.40 0.45 0.45 0.50 0.50 0.525 0.525 0.550 0.550 0.60 0.60 0.65 0.65 0.675 0.70 0.70 0.70 0.70 0.70 0.70 0.75 0.80 0.85 0.90 0.95 0.975 0.975 0.975 0.975

Below we show steps in the computation of the manifold of single tumble periodic motions, for fixed k and damping. This is a 2-manifold, and we plot it as period vs the two torques.

NEW The surface above were computed using the "old" algorithm. Here's a version computed using the "new" version:

Here are a set of high period (T=50) solutions at various values of the two torques, at fixed values of the spring constant k and the damping.
0., 0., 0., 0.05, 0.05, 0.1, 0.1, 0.1, 0.15, 0.15, 0.2, 0.2, 0.25, 0.25, 0.3, 0.3, 0.35, 0.35, 0.4, 0.4, 0.45, 0.45, 0.5, 0.5, 0.525, 0.525, 0.525, 0.55, 0.55, 0.55, 0.6, 0.6, 0.6, 0.65, 0.65, 0.65, 0.675, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 0.975, 1.0

Figure 5.5
Computed nearly homoclinic periodic motions as a function of the two torques for k=.1, damping=.5. Motions which tumble once per period are marked with a "+", those with two a diamond.

And here are a set of high period (T=50) single tumble solutions computed as a function of the two torques, at various values of the spring constant k and fixed damping. These are meant to approximate the curves of homoclinic connections.
0.3, 0.32, 0.33, 0.34, 0.35, 0.4, 0.43, 0.45, 0.4, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95

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