#foundations

Topics tagged "foundations".

• Foundations

  The core vocabulary of dynamical systems — states, evolution rules, and the geometry of how systems change over time.

• Graph

  A set of nodes joined by edges — the minimal mathematical object for describing relationships of any kind.

• Dynamical System

  A state together with a fixed rule that determines how that state evolves in time.

• State Space

  The space of all possible states of a system, in which each point is a complete snapshot and motion traces a trajectory.

• Phase Portrait

  The qualitative picture of all trajectories in state space, revealing a system's behavior at a glance.

• Fixed Point

  A state where the dynamics stand still — an equilibrium — classified by how nearby trajectories behave around it.

• Stability

  Whether small disturbances to an equilibrium grow or decay — the difference between a valley and a hilltop.

• Attractor

  The set a system settles onto after transients die away — a point, a loop, or something stranger.

• Bifurcation

  A qualitative change in a system's behavior as a parameter crosses a critical value — equilibria born, destroyed, or swapping stability.

• Lyapunov Exponent

  The average rate at which nearby trajectories pull apart — the quantitative fingerprint of chaos.

• Flows and Maps

  The two ways time can pass — continuously, as a flow of differential equations, or in discrete steps, as an iterated map.