---
tag: 116TPXR5FS
kind: concept
name: Smooth dynamical system
msc: [37C05, 37C10]
specializes: [BBZP1R872N]
sources:
  - url: https://en.wikipedia.org/wiki/Dynamical_system
---

## Summary

A smooth dynamical system is a [[dynamical system|BBZP1R872N]] whose phase space is a [[smooth manifold|F45VQ47GPV]] and whose time evolution is differentiable.
In discrete time it is a [[diffeomorphism|0KZZ70ZHXP]] iterated forward and backward; in continuous time it is the [[flow|06738GGWWK]] of a smooth vector field.
Differentiability allows one to linearise the motion along orbits, which leads to hyperbolicity, invariant manifolds and Lyapunov exponents.
Most dynamical systems of mechanics and of the sciences are smooth.

## Explanations

### L1 human

A smooth dynamical system is a rule that moves the points of a smooth space, such as a surface, without tearing or creasing it.
Either the rule is applied in steps, or the points flow continuously along arrows drawn on the space.
Near each orbit the motion looks, to first order, like a linear map, so tools from linear algebra apply.
For example, rotating the circle by a fixed angle, or letting a pendulum swing, are smooth dynamical systems.
See the [[definition|HZKFNZTHZS]].