Discrete-time dynamical system

Mathematical statement

A discrete-time dynamical system consists of a state space X and a self-map T:X→X. From x_0∈X, the recurrence x_(n+1)=T(x_n) defines a forward orbit. This slice iterates only complete explicitly given finite state spaces.

Representative formula (exact TeX)

x_{n+1}=T(x_n)

Assumptions

• Time is indexed by natural numbers including 0, and T is a total self-map on X.
• The question bank uses finite residue-class states and affine updates; a finite prefix implies no continuous-time model, asymptotic stability or chaos.

Symbol scope

• x — state orbit
• n — discrete time index
• T — state-update self-map

Intuition, definition, and misconceptions

The initial state and update map jointly determine the orbit. A state set without T does not determine evolution.

A fixed point satisfies T(x)=x, while a forward-invariant subset A satisfies T(A)⊆A. Finite state spaces permit complete enumeration of these properties.

A finite orbit or a cycle in a finite state space is not a continuous-time model and proves no general parameter-family stability, periodic classification or chaos.

Do not present a few iterates as a general long-term theorem or read a discrete update directly as a continuous differential equation.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Set — The state space is represented by a set.
• Function mapping — The update rule is a total self-map on the state space.
• Natural number — Natural numbers index discrete time.

Verification boundary

This slice checks only updates, orbit prefixes, fixed points and invariant subsets of an explicit finite self-map; it asserts no general dynamical-systems theorem.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Develop layered definitions and formalization of orbits, periodic points, attraction and stability.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.