Abstract
The basics of dynamical systems are reviewed: phase plane analysis and bifurcation theory. The idea of a strange attractor for ordinary differential equations is introduced, the phenomenon of fluid turbulence is discussed and the subject of stochasticity in Hamiltonian systems, and particularly in celestial mechanics, is described. The chapter finishes with a discussion of the Lorenz equations, the derivation of the Lorenz map and a simple characterisation of chaos in one-dimensional maps.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A nonlinear oscillator is an equation of the form \(\ddot{x}+V'(x)=0\); the quantity V(x) is called the potential (energy) on account of its rôle in the energy-like conservation law \(E=\tfrac{1}{2}\dot{x}^2+V(x)\).
- 2.
We define \([x]_+=\max (x, 0)\).
- 3.
In a nutshell: if you know precisely the state of the universe at time \(t_{0}\), together with the deterministic laws governing it, then (if you’re God) you will know the state of the future. At an atomic level, this is countermanded by the uncertainty principle, and, at a practical macroscopic level, by the subject of this book.
- 4.
Now apparently eight since the relegation of Pluto.
- 5.
Currently thought to be about ten days.
- 6.
The Boussinesq approximation neglects density variations except in the buoyancy term. It is equivalent to the assumption that \(\alpha \Delta T \ll 1\), where \(\alpha \) is thermal expansion, and \(\Delta T\) the prescribed temperature drop.
- 7.
A set M is dense in X if X is the closure of M.
- 8.
We will have plenty of discussion of manifolds later, but essentially in \({\mathbf{R}}^n\) they are simply subspaces.
- 9.
Period-doubling is discussed in Chap. 2.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Fowler, A., McGuinness, M. (2019). Introduction. In: Chaos. Springer, Cham. https://doi.org/10.1007/978-3-030-32538-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-32538-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-32537-4
Online ISBN: 978-3-030-32538-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)