Is π a Chaos Generator?

We consider a circular motion problem related to blind search in confined space. A particle moves in a unit circle in discrete time to find the escape channel and leave the circle through it. We first explain how the exit time depends on the initial position of the particle when the channel width is fixed. We then investigate how narrowing the channel moves the system from discrete changes in the exit time to the ultimate ‘countable chaos’ state that arises in the problem when the channel width becomes infinitely small. It will be shown in the paper that inherent randomness exists in the problem due to the nature of circular motion as the number 𝜋 acts as a random number generator in the system. Randomness of the decimal digits of 𝜋 results in sensitive dependence on initial conditions in the system with an infinitely narrow channel, and we argue that even a simple linear dynamical system can exhibit features of chaotic behaviour, provided that the system has inherent noise.