There's no solution in the same way that there's no "solution" to winning rock-paper-scissors. The cycle is endless because the desire to be in control is a key part of human nature, whether that be an authoritarian "I want everyone to do what I say" or a more oligarchic "I accept that there's others at my level, so we can cooperate so that everyone else does what we say", and any attempt to change those systems requires an equivalent amount of force that can all too easily lead one into side-tangents of trying to keep said force focused.

As a side note, Machiavelli identified the cycle in politics in his "Discourse on Livy" - a powerful and strong-willed individual takes power (e.g. Caesar or Napoleon), his descendants wield power with less and less efficiency until in time the aristocracy seize the reins, and they get more and more corrupt and out of touch until finally the people rise up and enforce some level of democratic sway. Unfortunately, the price of freedom is eternal vigilance, which is exhausting, and so over time things run down until some powerful and strong-willed individual takes power and it all starts again. It's not purely linear - an aristocracy can be subsumed into a strong individual leadership (e.g. the popes in the 19th century grabbing power back from the cardinals) and a king can be overthrown by a democratic uprising (e.g. Louis XVI of France - though technically it did go through a brief aristocratic moment, as he re-convened the parliament to try and get around the nobility who wouldn't fund his wars, indicating his powers had weakened). But in general we oscillate between these three modes of social organisation because of the difficulty in centralising power and in then keeping it from being corrupted (i.e. using it for selfish purposes) once it is centralised.