Alright, it shows some personal quality of yours that you can take criticism (especially if formulated more harshly than was necessary when I read it in restrospect) and trying to address it. Did not think this would happen but it shows your sincerity and openness.
If you really believe you are onto something, I would advice you to also have a look at complexity theory and emergent properties of complex systems. Complex systems with many individual parts, each one following a different set of simple rules, are known to (at least sometimes) create systems that are more than the sum of their individual parts.
The only way I can imagine your pendulum thought-construct to hold some value is if
A) one considers society dynamics as an emergent property of the interactions of millions/billions of individual actors
B) where there are at least two discrete states (compared to continuous) of a society that are stable (local optima), whereas all other continuous states in between are unstable.
C) these two discrete states of a society correspond with your right/left brain characteristics
In that case, you would have the burden of proof to show that discrete states in societies exists, that there are only 2 optima (and not 1, 5 or 17) where societies fall into and are stable for a certain amount of time. Furthermore, you would have to define aforementioned parameters of how you want to meassure societal stability and then look at the historical data. Then you would need to estimate correction parameters for incomplete datasets (with happens often in historical data) and do everything else possible to challenge potential biases.
If you have done all of this, you would only know (with a defined degree of certainty) that discrete societal states exists and that they can change from one to the other, but you would still be oblivious to the causes.
To estimate causes driving societal state transition you would have to provide serious data on the human condition/biology/sociology I am not aware of if they even exists or can exist. Let’s say you would argue it is a collective mind change from left to right brain; what are the molecular mechanisms behind this change? Is it genetically embedded in our DNA? (Not likely since it’s a transgenerational and trans-relatedness phenomenon). Is it environmental conditions (maybe left-driven societies always create an environment of inequality, and exposure to inequality shapes brain development in children)… How do you account for factors effecting brain development that none of our ancestors were exposed to? Social media and dopamine system comes to my mind…
Like with any scientific endeavor, one has to start somewhere; so I would advice you to first hammer it down with facts that indeed 2 discrete states of society exists and that these can be discribed with left/right brain characteristics as estimated by as many independet parameters as possible to avoid confounding.
Good luck! Keep us updated!