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What is the qualitative explanation for why a higher driving amplitude lead to chaos in a driven pendulum?

stafusa
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Ee Kin Chan
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To be in same page: Being chaotic means, e.g., that an arbitrarily small change can dramatically influence the system's evolution.

I think we can interpret this question in two ways:

1. How can high driving amplitudes lead to chaos?

The most intuitive explanation I can thing of for this specific system is the following:

  • At its unstable equilibrium, standing vertically upwards, any tiny difference in speed or position can change whether it'll move left or right. So, if the driving force is strong enough to always bring the pendulum back to its unstable equilibrium at low speeds, then the system will recurrently find itself in this "sensitive" point, leading it to exhibit the sensitivity to initial conditions characteristic of chaos.

The reason it shouldn't arrive too fast at that point is that the instability effect could then be lost and the dynamics completely dominated by the forcing.

One could perhaps also simply say that chaos is ubiquitous among nonlinear systems, and a strong enough forcing can make the pendulum nonlinear enough to display chaos, but I guess few would find this satisfying. Related: Physical reasons for why systems are chaotic?.

2. Why can't arbitrarily small driving amplitudes lead to chaos?

The model will usually include damping, so that the pendulum's energy can't diverge. So, if the forcing amplitude $A$ is not high enough for it to compensate damping loses even in a resonant mode, then the dynamics unavoidably converges to small amplitude oscillations around the bottom stable equilibrium point, which is a regular attractor, and you don't get chaos. Therefore, a hard lower bound for the forcing amplitude is given by the damping factor — $A$ cannot be arbitrarily small.

Another, more mathematical way to see this is to remember that if the pendulum is restricted to move around its stable equilibrium point, its behavior gets close to linear, and linear systems don't display chaos — the pendulum will tend to oscillate around $(0,0)$ at approximately the driving frequency.

If one is interested in chaos generically, we have our recommended books and a plethora of online resources, such as this or this, to give two recent, open access examples.

stafusa
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