dynamical systems theory

Dynamical systems theory, also known as the qualitative theory of differential equations, is a branch of mathematics that studies the general properties of solutions to differential equations without explicitly solving them. It was pioneered by Henri Poincaré in the late 19th century, partly in response to a prize offered for determining the stability of the solar system.1

This theory combines local analytic information with global geometric and topological properties of the "state space" or "phase space," which represents all possible solutions. A key discovery within this field is "chaos," which describes systems governed by deterministic laws that exhibit apparently random or unpredictable behavior due to extreme sensitivity to initial conditions. This concept, famously illustrated by the "butterfly effect," challenges the idea of a purely predictable "clockwork universe". Chaos theory has found applications in diverse fields, including weather forecasting, fluid dynamics, and population dynamics.12