Abstract
Continuous systems with only one or two variables cannot be chaotic. However, such systems are used frequently in pretty much all scientific fields as “toy models” that capture the basic relations of certain variables, so it is useful to know how to handle them. Although these non-chaotic systems can be treated analytically to some extend, this is always context-specific. Here we will show how to understand them generally and visually, by looking at their state space. An important tool for performing such a graphical analysis are so-called nullclines, the use of which is demonstrated with an excitable system that exhibits a high amplitude response when perturbed above a relatively small threshold. We also discuss the mental steps one takes while creating a model, and best practices for “preparing” the model equations for study within dynamical systems theory context. Quasiperiodicity is a typical form of non-chaotic motion in higher-dimensional dynamics, and thus serves as a fitting end to this chapter.
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.
In Sect. 1.4 we discussed the criteria for stability based on the eigenvalues of the Jacobian. Here, since there is only one dimension, the Jacobian is a number, \(J\equiv df/dx\), and thus its eigenvalue coincides with itself.
- 2.
Here inwards means that the dot product of f with the normal vector of R at any point \(\mathbf {x}\in \partial R\) is negative.
- 3.
Consider \(\begin{aligned} & u(t_{0}+\Delta t) - u(t_{0} - \Delta t) = \int _{t_{0} - \Delta t}^{t_{0}+\Delta t} \dot{u} dt = \int _{t_{0} - \Delta t}^{t_{0}+\Delta t} au(u-b)(1-u) dt +\int _{t_{0} - \Delta t}^{t_{0}+\Delta t} w\, dt \\ & +\int _{t_{0} - \Delta t}^{t_{0}+\Delta t} I_{0} \delta (t- t_{0}) dt \\ \end {aligned}\) which goes to \(I_0\) for \(\Delta t \rightarrow 0\).
- 4.
The real numbers \(\omega _1, \omega _2, \ldots , \omega _k\) are rationally independent if the only k-tuple of integers \(m_1, m_2, \ldots , m_k \in \mathbb {Z}\) that satisfy the equation \(m_1 \omega _1 + m_2 \omega _2 + \cdots + m_k \omega _k = 0\) is the trivial case of \(m_i = 0 \, \forall i\).
- 5.
A k-torus (short for k-dimensional torus) is the Cartesian product space of k circles. It is a manifold where each of its dimensions is given by an angle.
- 6.
The Lotka-Volterra is unrealistic because it provides oscillations of arbitrary amplitude depending on initial condition. In reality, while predator and prey populations do follow lagged oscillations with each other’s population, there is typically a single characteristic amplitude (limit cycle) that describes the dynamics [30].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Datseris, G., Parlitz, U. (2022). Non-chaotic Continuous Dynamics. In: Nonlinear Dynamics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-91032-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-91032-7_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-91031-0
Online ISBN: 978-3-030-91032-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)