Dynamical systems theory

Dynamical systems theory studies systems that change over time according to specified rules. It analyzes their trajectories, stability, and long-term behavior.

Phase space: A space whose points represent all possible states of a system. A trajectory is a path through this space as the system evolves.
Ordinary differential equation: An equation relating a function of one variable to its derivatives. Continuous-time systems are often specified by differential equations.
Henri Poincaré: A French mathematician whose work founded modern qualitative analysis of differential equations. His study of celestial mechanics introduced tools central to modern dynamics.
Lotka–Volterra equations: A pair of differential equations modeling interactions between predator and prey populations. Their cycles show how nonlinear interactions generate sustained population fluctuations.
Control theory: The study of steering dynamical systems toward desired states using feedback or inputs. It emphasizes purposeful intervention, while dynamical systems theory also studies unforced behavior.
Dynamical system: A rule specifying how a system's state changes over time. The theory studies these rules in discrete and continuous time.
Initial value problem: A differential equation together with a specified starting state. The starting state selects a particular trajectory from the system's possible solutions.
Isaac Newton: An English mathematician and physicist who formulated classical laws of motion and gravitation. Newtonian mechanics supplied an early model of rule-governed time evolution.
Lorenz system: A three-dimensional system of differential equations with chaotic solutions. It arose in a simplified model of atmospheric convection and illustrates sensitive dependence.
Ergodic theory: The study of measure-preserving transformations and statistical properties of dynamical systems. It asks how long-term time averages relate to averages across state space.

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