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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

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

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

we define the flow by [1]

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


next up previous contents
Next: Linear systems Up: Theory Previous: Theory   Contents
Antti Honkela 2001-05-30