Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Abstract Category of Dynamical Systems

James Schmidt

Abstract

A notion of time is fundamental in the study of dynamical systems. Time arises as a standalone dynamical system and also in solutions or trajectories as a special kind of map between systems. We characterize time by a universal property and use universality to motivate an abstract definition for categories of dynamical systems. We propose this definition as guidance in concrete instantiations for other kinds of systems.

An Abstract Category of Dynamical Systems

Abstract

A notion of time is fundamental in the study of dynamical systems. Time arises as a standalone dynamical system and also in solutions or trajectories as a special kind of map between systems. We characterize time by a universal property and use universality to motivate an abstract definition for categories of dynamical systems. We propose this definition as guidance in concrete instantiations for other kinds of systems.
Paper Structure (7 sections, 3 theorems, 7 equations)

This paper contains 7 sections, 3 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Ordinary Dynamical Systems
  3. Continuous-Time Systems
  4. Complete Time is a Universal Complete Dynamical System
  5. Local Time is a Universal Continuous-Time Dynamical System
  6. Discrete Time is a Universal Discrete-Time Dynamical System
  7. Abstract Systems

Key Result

Theorem 1

Let $(M,X)$ be a dynamical system, and $x_0\in M$. Then there are $t_0<0<t_1\in \mathbb{R}$ for which smooth map $\varphi_{X,x_0}:(t_0,t_1)\rightarrow M$ is unique maximal solution of $(M,X)$ with initial condition $x_0$. Thus, $\varphi_{X,x_0}(0) = 0$ and $\frac{d}{dt} \varphi_{X,x_0}(t) = X(\varph

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Theorem 1
  • proof
  • Definition 2.5
  • ...and 22 more