Cuckoo hashing proof of cycle
Yes there is, using algorithms such as Brent's cycle finding algorithm.

However, this is quite expensive and has fairly complicated code. What most implementations do is define some constant cycle/chain length, for example $\log n$ where $n$ is the number of elements in the hash table. If you find a chain (or cycle, but we never investigate whether it's actually a long chain or a cycle) larger than that constant we rehash or increase the hash table size.

But if your goal is merely the prove that you have a cycle, and you do not care about efficiency, the simplest method is to invoke the pigeonhole principle, and just loop $n + 1$ times where $n$ is the number of possible slots in the hash table. If you never find an empty slot you must have a cycle.