period-doubling bifurcation

A period-doubling bifurcation occurs in a system where, as a parameter changes, a steady or simple repeating behavior becomes unstable and transitions into a more complex cycle that repeats every two loops instead of one. Imagine a pendulum that initially swings back and forth regularly; after the bifurcation, it starts swinging with a pattern that repeats after two swings, creating a pattern that doubles its period. This process can repeat, leading to increasingly complex behavior and often signals the onset of chaos in dynamic systems.