Have you ever heard of a feedback loop? If you haven't, I can best describe it as a way in which a system can actively detect its output through a sensor and make changes to its variables in order to achieve and maintain a certain desired state automatically.
Control theory is a branch of mathematics mainly used in Electrical, Automation and Robotics engineering, extremely useful for the autonomous operation of devices.
Example: control systems are extremely prevalent in the aerospace field, used to maintain the orientation of satellites, to autopilot planes and missles and to program robots used for space exploration.
But I'm not here to discuss aerospace (unless we're talking Breitling 😉). Here's how control theory can be applied to mechanical watchmaking, in the form of the resonance synchronization found in clocks and watches by Breguet, Janvier and Journe among others.
We can describe the coupling of the two balance wheels of a Resonance through this system of ordinary differential equations (for the sake of simplicity I'll just leave the definition of some of the variables for the answers to any eventual questions, just know that Theta describes the angular position of the balances, omega is the angular velocity of such balances and k signifies the rigidity of the base plate, through which the sound waves propagate in order for the coupling to happen).
As we all know, our beloved mechanical watches don't contain any electronics, so the way in which the oscillators control one another can be described as passive reciprocal control. This makes the system "open", which means that it lacks a feedback loop.
If we assume that the watch is kept wound, the oscillations of the balance wheels can be described through this graph: as you can see, the amplitude is kept constant and the oscillators are in phase opposition, in such a way they are able to compensate one another, thus stabilizing the system. (The simulation was performed using MATLAB and Simulink).
What I love about watchmaking is that it can be appreciated through many lenses, and science is one of them. Hopefully I was clear enough in my explanation. Feel free to ask any questions, I'll try to answer them if I can.