Dynamical systems and ergodic theory

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• ConceptDynamical system

  A dynamical system is a space of states together with a rule that moves each state forward in time. Time can be discrete, as in the repeated application of one map, or continuous, as in the solutions of an autonomous differential equation. The rule must be consistent: waiting for time $s$ and then for time $t$ is the same as waiting for time $s + t$. Dynamical systems model planets, populations, chemical reactions and many other processes, and their study asks what happens in the long run.

• ConceptFlow of a vector field

  A flow is a family of maps $\varphi_t$ of a space to itself, indexed by real time $t$. The map $\varphi_0$ is the identity, and $\varphi_s \circ \varphi_t = \varphi_{s+t}$. The flow of a vector field sends each point to the position, after time $t$, of the solution of $x' = f(x)$ through it. See the definition.

• ConceptDiscrete-time dynamical system

  A discrete-time dynamical system is a dynamical system in which time advances in whole steps. It is given by a single map $f$ of a space to itself, and the state after $n$ steps is the iterate $f^n(x)$. Such systems arise from sampling a continuous motion at regular times, from population models with generations, and from numerical methods. Many basic phenomena of dynamics, such as chaos and period doubling, are already visible in maps of an interval.

• ConceptFixed point

  A fixed point of a map is a point that the map sends to itself. In a dynamical system, a fixed point is a state that never changes, the discrete-time analogue of an equilibrium point. Many problems, such as solving an equation $g(x) = 0$, can be rewritten as finding a fixed point. Theorems such as the Banach fixed point theorem guarantee that one exists. Near a fixed point, the behaviour of the iterates tells whether nearby states are attracted or repelled.

• ConceptPeriodic point

  A periodic point of a map is a point that comes back to itself after some positive number of steps. Its orbit is a finite cycle that repeats forever. Fixed points are the periodic points that return after one step. Periodic points are the simplest kind of long-time behaviour, and counting them says a lot about a system. For flows the analogue is a periodic orbit.

• ConceptMinimal period

  The minimal period, or prime period, of a periodic point is the least number of steps after which it returns to itself. It equals the number of distinct points in its orbit. Every period of the point is a multiple of the minimal period. Results such as Sharkovskii's theorem are stated in terms of minimal periods.