motivating dynamical systems
What is “dynamics”? A dynamical system, in general, is a monoid action on a set.
Ignoring the generality, let’s consider the definition of a discrete dynamical system: a function
on a set is considered to be a discrete dynamical system (autonomous) where the iterations of the map
are considered the time map for . Thus orbit of a point is defined to be
Simply put, just “iterations of a map” produces the dynamics.
Using this simple, extremely general definition, we reinterpret a lot of math in terms of “dynamics”:
- Linear algebra is study of iterations of one linear map on a vector space. The canonical forms essentially decomposes the dynamics into indecomposable pieces.
- Finite group theory studies group action on a (finte) set are by definition “dynamics”
- A module of a ring is just a special group action by the additive group of the ring which also plays well with the multiplicative structure. Representation theory in most cases studies such modules.
- There are various applications of theorems that are “dynamical”
- proof of existence of ODEs using contraction mapping theorem
- Solutions of an ODE (with unique solutions) on produces a flow map
on the phase space that map initial conditions to solution curves (this is the definition). Thus this is a “continuous-time dynamics”.
- A common proof of existence of solutions to ODEs uses a fixed point theorem, which is in turn a “dynamical” theorem!
- Machines have a natural monoid action.