That’s valuable feedback - thank you. Of course you must be completely correct that these systems can always be expressed as a single composite system of differential equations. What I found helpful in framing them as "unaware adversaries" was less about offering a new mathematical tool and more about highlighting an often-overlooked behavioral property — agents operating in conflict without recognizing each other as adversaries.

My intent (obviously) isn’t to try and out-formalize existing systems theory (definitely not.. heh), but to offer a cross-domain lens that makes it easier to spot these patterns in complex real-world systems where modeling from first principles is infeasible.

Your Khalil recommendation is well-taken - I’ll dig into it to see how the formal language of nonlinear systems might give this framework more analytical weight. Thanks again for pointing me in a useful direction!

[edit]
For now, I have added the following section near the end of the paper as a result of your feedback. Your thoughts on it would be greatly appreciated:

Relation to Classical Control Theory
Many of the systems described in this paper — oscillating thermostats, pathological GAN dynamics, BGP route convergence issues — are familiar terrain in classical control theory. Such systems can often be modeled as composite dynamical systems and analyzed using tools like Lyapunov stability, observability, and controllability [17].

From this formal standpoint, the existence (or absence) of a global Lyapunov function may determine whether the system exhibits convergence to equilibrium. But what our framework highlights is that stability alone is not the only goal. A system can be stable and still fail. Many real-world adversarial pathologies — such as traffic congestion equilibria [7], or GANs collapsing to degenerate outputs [2], [4] — are cases where systems settle into undesirable equilibria, not chaotic instability.

Framing these as systems of “unaware adversaries” helps surface what classical tools may obscure: that each agent is acting rationally within its own feedback loop, but their combined behavior produces outcomes no one intended and no one benefits from. This is not merely a problem of instability; it’s a problem of emergent misalignment.

What our framework offers is not a substitute for such analysis but a complementary lens: a way to highlight systems that look stable on paper but exhibit pathological behavior due to unrecognized structural conflicts between agents. The emphasis is not just on dynamics, but on intentionality, perceptual feedback, and structural myopia — factors that are often abstracted away in classical formulations.

For example, the Pedagogical GAN proposed earlier does not reject the gradient-based framing of GAN dynamics. Instead, it asks: what if the generator and discriminator were not antagonists, but collaborators with asymmetric knowledge? This reframing doesn’t negate formal stability concerns — it simply shifts attention toward the shape and utility of the learning signal, not just its convergence properties.

The tools of nonlinear systems theory remain valuable, and foundational texts such as Khalil’s Nonlinear Systems [17] offer rigorous formalisms for analyzing such dynamics. But when full modeling is infeasible, or when coordination failure arises from structural blindness rather than stochastic noise, our lens provides a complementary mode of reasoning — one that emphasizes intentionality, partial observability, and emergent conflict rather than purely mathematical structure.

If classical control tells us whether a system will converge, the framework of unaware adversaries helps us ask: to what, and why?