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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

 (2.1)

where and is a diffeomorphism, i.e. a smooth mapping with a smooth inverse. It is important that the mapping is independent of time, meaning that it only depends on the argument point . Such mappings are often generated by flows of autonomous differential equations.

For a general autonomous differential equation

 (2.2)

we define the flow by [1]

 (2.3)

where is the unique solution of Equation (2.2) with the initial condition , evaluated at time . The function in Equation (2.2) is called the vector field corresponding to the flow .

Setting , where , 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 . Thus it is a discretisation of the continuous system.

Next: Linear systems Up: Theory Previous: Theory   Contents
Antti Honkela 2001-05-30