# Discrete-time dynamical system

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability.

mko-discrete-dynamical-system · definition · v0.10.0



## 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

[Discrete-time dynamical system: iterating T(x)=(3x+1) mod 7 on seven states](https://ocme.evemisslab.com/en/derivations/derivation-mko-discrete-dynamical-system-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — The state space is represented by a set.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The update rule is a total self-map on the state space.
- [Natural number](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/discrete-dynamical-system.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-discrete-dynamical-system/index.md)