Dynamical Systems and Differential Equations

General concepts and structures studied by dynamical systems theory, the branch that describes how a state evolves over time under a fixed rule. Belongs here: the Dynamical System itself, the pairing of a state space with a rule, discrete or continuous, that determines every later state from an earlier one; the Phase Space, the set of every possible state a system can occupy, whose geometry is what a dynamical system actually studies; the Attractor, a set of states a system settles toward from a range of starting points, the long-run behavior the field is usually most interested in; Bifurcation Theory, the study of how a system's qualitative long-run behavior changes abruptly as a parameter is varied smoothly. Does not belong here: the branch entity Dynamical Systems and Differential Equations itself, and named individual mathematical objects belonging to this cluster, which are typed as mathematical-object rather than concept even when they use these same underlying structures.