Even as a single pendulum there's a huge number of possible trajectories it could take, making it effectively random from our perspective.

A simple pendulum is actually a chaotic system in the real world (everything is, for the most part). It's extremely sensitive to initial conditions. Even microscopically small differences in how hard the cat swipes, the exact angle of impact, tiny air currents, or imperceptible vibrations in your hand holding the toy all get amplified exponentially over time into completely different swing patterns.

The phase space is basically the map of all possible positions and velocities the toy could have. It contains an astronomically large number of distinct states it can pass through. We're talking numbers so large they're bounded only by physical limits like the Bekenstein bound. The path through these states depends on initial conditions we simply cannot measure precisely enough, plus constant perturbations from the environment (thermal fluctuations, air resistance, microscopic imperfections in the string).

Add to this that cat toy strings aren't perfectly flexible. They have some rigidity and internal friction that creates additional complexity and damping. This makes the system even less predictable because the energy dissipates in chaotic ways.

Whether the underlying universe is truly random (quantum indeterminacy) or just deterministically chaotic beyond our ability to measure doesn't matter for practical purposes. The computational precision required to predict the trajectory exceeds what's physically possible to measure.

For all intents and purposes, every swat creates a genuinely unpredictable and effectively random experience for the cat.