Dynamical systems

The theory of dynamical systems is the basic mathematical tool for analysing time series. This section presents a brief introduction to the basic concepts. For a more extensive treatment, see for example [1].

The general form for an autonomous discrete-time dynamical system is the map

$$\mathbf{x}_{n+1} = \mathbf{f}( \mathbf{x}_n )$$ (2.1)

where $\mathbf{x}_n, \mathbf{x}_{n+1} \in \mathbb{R}^n$ and $\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a diffeomorphism, i.e. a smooth mapping with a smooth inverse. It is important that the mapping $\mathbf{f}$ is independent of time, meaning that it only depends on the argument point $\mathbf{x}_n$. Such mappings are often generated by flows of autonomous differential equations.

For a general autonomous differential equation

$$\mathbf{x}'(t) = \mathbf{f}(\mathbf{x}(t)),$$ (2.2)

we define the flow by [1]

$$\phi^t(\mathbf{x}_0) := \mathbf{x}(t)$$ (2.3)

where $\mathbf{x}(t)$ is the unique solution of Equation (2.2) with the initial condition $\mathbf{x}(0) = \mathbf{x}_0$, evaluated at time $t$. The function $\mathbf{f}$ in Equation (2.2) is called the vector field corresponding to the flow $\phi$.

Setting $g(\mathbf{x}) := \phi^\tau (\mathbf{x})$, where $\tau > 0$, gives an autonomous discrete-time dynamical system like in Equation (2.1). The discrete system defined in this way samples the values of the continuous system at constant intervals $\tau$. Thus it is a discretisation of the continuous system.