Abstract
Problems that can be written as singularly perturbed systems of first order differential equations can be solved using approaches that combine matched asymptotic expansions with phase plane analysis. The \(\varepsilon \rightarrow 0\) limit yields a separation of time-scales that reduces the overall system to different forms for the fast and slow dynamics over different intervals of time. Asymptotic matching is used to connect the fast and slow solutions and can be interpreted in terms of the geometric structure of nullclines and the structure of the phase plane.
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.
Attempting to put this equation in the form (9.2) fails because dividing by \(\varepsilon \) suggests a very fast oscillation, \(\omega _0=\varepsilon ^{-1/2}\rightarrow \infty \), and a large (\(O(\varepsilon ^{-1})\rightarrow \infty \)) rather than small perturbation term.
- 2.
At which point, the solution is no longer described by the dynamics of the slow system.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Witelski, T., Bowen, M. (2015). Fast/slow Dynamical Systems. In: Methods of Mathematical Modelling. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23042-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-23042-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23041-2
Online ISBN: 978-3-319-23042-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)