Introduction

In dynamical systems theory it is a common problem to analyze a discrete set of observed quantities, a time series. Consider for example the dynamical system defined by the differential equation

$$\quad \frac{dy}{dt} = F(y) \quad,$$ (1)

where $y=y(t)=\left( y_{1},y_{2},\ldots,y_s\right)^T \in S$ is a vector representing the state of the system at time $t$ in some $s$-dimensional phase space $S$. The vector field $F:S\to TS$ ($TS$ is the tangent space of $S$.) induces the flow $\varphi_{t}:S\to S$ :

$$\quad y(t) = \varphi_{t} (y_{0}) \quad;$$ (2)

here, $y_{0}$ comes from the initial condition $y(t=0)=y_0$. Often, when one analyzes a system, the analytic solution $y(t)$ is not known. Instead, only a time series is available, a series of values of one single quantity sampled at regular intervals:

$$\quad v(y_0),v(\varphi_{\tau}(y_0)),v(\varphi_{2\tau}(y_0)),\ldots \ \equiv \ v_0,v_1,v_2,\ldots \quad.$$ (3)

$v(\varphi_t(y_0))=v(y(t))$ is the value of some observable $v$ of the system at time t. The time interval $\tau$ between two successive measurements of $v$ is the sampling time or delay time. $\tau$ need not be fixed; it is possible to consider the sequence

$$v(\varphi_{t_0}(y_0)),v(\varphi_{t_1}(y_0)),v(\varphi_{t_2}(y_0)),\dots \equiv v_0,v_1,v_2,\dots \quad,$$ (4)

as well, where the times $t_0<t_1<t_2<\cdots$ are not equidistant. One can get such a time series as the result of an experiment (where it is often convenient to measure only one single observable) or from the numerical integration of eq. (1). It is clear that having only a time series, instead of the complete solution, is a strong restriction of our knowledge about the system. Alternatively, one might have a discrete dynamical system, defined by the mapping

$$\quad y_{n+1} = \varphi(y_n), \ \ \ \ n=0,1,2,\ldots \quad,$$ (5)

where $y_n\in S, \ \varphi:S\to S$, and get the time series $\{v_n \equiv v(y_n)\}_{n\in\{0,1,2,\ldots\}}$. As the system evolves, the trajectory in the phase space $S$ approaches an attractor¹ $A$ which lies within some submanifold $M$ of $S$:

$$\quad A \subseteq M \subseteq S \quad,$$ (6)

where

$$\quad \dim A \leq \dim M \leq \dim S \quad.$$ (7)

The general aim of the analysis of the time series is to describe the behaviour of the system on this attractor $A$. One would like to reconstruct a geometric picture of $A$ in phase space. Also one is interested in characterizing $A$ by calculating its Hausdorff-Besikovich (or fractal) dimension, information dimension, correlation dimension, Lyapunov exponents, etc. The problems one encounters when computing these quantities from a time series will be discussed in this essay. In section 2, we will show how an embedding trick can be used to get a simple technique of reconstruction of phase space pictures. Section 3 deals with the application of this technique. In particular, we will present several strategies how to choose some of the important parameters of the reconstruction. Additionally, we will consider the problems arising from noisy data, and we will discuss the requirements the data (i.e. the time series) has to fulfill to allow for reasonable results. The important news is that, in fact, it is possible to get those reasonable results, despite of the very restricted form of the data.

Footnotes

...attractor¹

We assume that our dynamical system is dissipative, i.e. the phase flow $\varphi_t$ (or the map $\varphi$, respectively) contracts volume in (some part of) the phase space. Otherwise the discussion of attractors would not be very meaningful.