math for things that both flow like water and flip like light switches.
means a hybrid system is a dynamical system that mixes continuous behavior, like motion or temperature, with discrete jumps, like a mode switching or a collision.
from emerged in the 1990s from control theory and computer science colliding, as engineers realized real machines like thermostats and robots don't just evolve smoothly, they also make sudden logical decisions, so continuous calculus needed discrete automata bolted on.
thermostat control — switches heater on and off around a temperature threshold
aircraft autopilot — blends continuous flight dynamics with discrete mode changes like landing gear
traffic light networks — discrete signal states governing continuous vehicle flow citywide
cardiac pacemakers — jumps between pacing modes based on continuously sensed heart signals