Coupled pendula — that is, two or more pendula attached to one another — are a system that is well known to be chaotic for large amplitude oscillations. Less well known is the interaction that occurs between resonant coupled pendula at small amplitudes of oscillation which, unlike chaotic motion, leads to periodic behaviour and a period of motion much greater than that of either pendulum alone.
Rott's pendulum is the name given to a pair of coupled pendula with two properties: firstly that the pivots of the two pendula are horizontal when the system is in stable equilibrium, and secondly that the resonant frequencies of the pendula are close to an integer ratio (usually 2:1). The dynamics of this system were first analysed by Nikolaus Rott (Journal Zeitschrift für Angewandte Mathematik und Physik, 21 (4), 1970, 570-582), who demonstrated also that such a pendulum can be modelled with rigid rods of constant mass per unit length in the following configuration:
For the red pendulum to have a frequency of λ times that of the black one, it can be shown that the ratio of the length α of the horizontal rod to the unit length of the vertical rods must be the real root of the equation:
Taking the frequency ratio λ=2 gives the solution α ≈ 1.2835670
Taking the system to be frictionless, the (dimensionless) equations of motion can be shown via construction of a Lagrangian to be:
where Δθ = θ1 − θ2. This fourth-order system of differential equations is analytically intractable as it stands, although easy to solve numerically. If a simplifying assumption is made, that θ1, θ2, θ'1 and θ'2 are all small quantities of order ε and that terms of order ε2 can be neglected, the equations become simply:
The key feature of this simplification is that the two equations now each only involve one variable; this a direct consequence of the condition that the pivots of the two pendula are horizontal when the system is in stable equilibrium. Physically this implies that, to first order, the motions of the two pendula are independent of each other, and that each swings with a constant (small) amplitude, as determined by the initial conditions. It is usually expected that as ε → 0, the behaviour of the full system tends to the behaviour of the simplified system, as the terms of order ε2 become decreasingly important.
However, for pendula with a frequency ratio λ ≈ 2, the assumption that terms of order ε2 can be neglected is not a good one. The simplified system forms two undamped oscillators. When the pendula have resonant frequencies a factor of two apart, some of the neglected terms of order ε2 in each equation are oscillatory, and of the same frequency as the linear oscillators. While the neglected terms are small, over a long period of time they will resonate with the oscillators and alter the solution to a much greater extent than their small magnitude would suggest. This is the same principle as pushing a playground swing: a small push applied repeatedly over time at the correct frequency causes the swing to go much higher than a single push would.
In the case of Rott's pendula, this mechanism results in slow energy exchange between the pendula: first the red pendulum swings and the black one is stationary, but over the course of many oscillations, the oscillation switches to the black pendulum. Conversely, as the black pendulum swings, it transfers energy to the red pendulum, and the system slowly returns to its starting state, with the black pendulum stationary and the red one swinging. This behaviour is visible in a simulation of the full, unsimplified equations.
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While the analysis of the nonlinear resonance between the pendula is valid only in the limit of ε → 0, the behaviour of the full system is characterised by the slow, periodic exchange of energy between the pendula for surprisingly large amplitude oscillations. However, for very large amplitudes of oscillation, especially when the pendula have enough energy to overturn, the behaviour becomes chaotic.
Behaviour of Rott's pendulum as a function of initial conditions. Hue and brightness show the amount of time required for the black and red pendula to first overturn, as a function of initial stationary position.
The white areas in the corners of the diagram represent the initial conditions where neither pendulum overturns before a cutoff time. In much of this region, there is insufficient energy for either pendulum to ever overturn, and the system remains in the mode of slow energy exchange indefinitely. The complex shape of the coloured region, where there is enough energy for one or both pendulums to overturn, is due to the high sensitivity of the system to initial conditions. Here, a very small change in the initial angle of the pendulum can lead to very different future behaviour, one of the defining features of chaotic dynamical systems.