Background material

Discrete dynamic systems

Dynamic systems are often described by differential equations. These equations model the relationships between a physical system's changes and its state. In order to understand a physical system's development in time, it is necessary to be able to integrate the system, i.e. to find solutions for the differential equations, either analytically as compact formulas or numerical by extensive use of computers.

One example of a dynamic system is this model taken from ecology. We consider two species that compete for space in a geographically limited area. The one species lives by eating the other. We let x=x(t) be the population of the predator, while y=y(t) is the population of the prey, both at time t. We can now set up a system of differential equations that describes the trend of the two populations:

This gives us a dynamic system. The solutions to the system will describe the trend of the two animal populations with time.

In a discrete dynamic system, the time parameter is converted to a discrete quantity. The equations that are included in the system describe the next state as a definite function of the previous state. A simple example is the discrete logistical growth model, given by:

Let us assume that x₀=0.5 and see what kind of development we get. In the table, we let the constant k assume four different values. We get four different trends; for k=1.5 we get a so-called fixed point, i.e. x_n gradually stabilises at a fixed value, in this case 0.3333... . For k=3.2 we get a different state of equilibrium, two different values for x_n and regular alternation between them. For k=3.5 we get something similar, but this time we alternate among four values. In all of these examples, we observe an attractor, or a set that the system approaches with time. For k=3.9, however, we get a completely new situation, the trend becomes chaotic, apparently without any easily recognisable system.

The logistical growth model is an example of a one-dimensional dynamic system. The Hénon map is an example of a two-dimensional system.

Like the one-dimensional system, this system also has fixed points, i.e. values where x_n+1=x_n and y_n+1=y_n. We find these at:

In Hénon's standard example, the values of the constants are a=1,4 and b=0,3. For the one fixed point, this gives x_n ≈ 0,63135 og y_n ≈ 0,18941. The Hénon map has a strange attractor (proved by Carleson and Benedicks), a set that is such that once the system has entered the attractor, it will remain there, but inside the attractor we have a chaotic trend.

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