Discrete-time systems

Often a physical setting is reduced to a set of measurements, for example, temperature, pressure, stock market prices, etc. In discrete-systems, we give these measurements at a sequence of specific times. We would hope that given the measurements at time n that we have a rule to determine the measurements at time n+1. If represents the measurements at time n, this rule may take the form

where f(x) is a given function fixed for all time. The evolution of the system is then obtained by iterating the function

A classic example studied intensively in the seventies is that of the quadratic functions f(x)=cx(1-x) where c is a fixed real number. The graph of this function is a parabola passing through the x-axis at x=0,1. The maximum value is c/4 occurring at x=.5. An example is shown below:

  
Figure 1: Graph of y=2x(1-x)

If we restrict c to be nonnegative and less than or equal to 4, the function f(x) has the property that it maps all points in the unit interval [0,1] to values in [0,1]. Such a mapping is called an interval self-mapping. We may narrow our attention to measurements inside [0,1].

Specific values of the measurements of the physical setting are often called the state of the dynamical system. In our example, the state space is the unit interval. If we start with an initial state of and fix c=2, it is easy to compute the subsequent states by means of the equation . The results are shown in the table below.

 

Table 1: Iteration of f(x)=2x(1-x) starting with

	0.1
	0.18
	0.2952
	0.41611392
	0.4859262512
	0.4996038592
	0.4999996862
	0.5000000000

 

We may easily guess the long-term behavior of this system: the limit of is .5 as . In fact, x=.5 has a special property with respect to this dynamical system; it satisfies f(.5)=.5, and thus qualifies as a fixed point of f(x), or a point of equilibrium of the dynamical system. A little experimentation will show that any initial state in (0,1) eventually leads to x=.5 in the limit. Thus, x=.5 is an example of a stable or attractive fixed point.

The advent of high-speed computers made possible a huge volume of experimentation in studying the behavior of dynamical systems. The numerical study of simple cases such as this interval mapping resulted in the discovery of fantastic new patterns in nature and gave rise to an explosively growing area of mathematical research. In particular, for higher values of c the behavior is nowhere near as simple as in the case c=2 we have mentioned. We will study the basic theory of this interval self-mapping later in the course.

One theme in dynamics is the use of symbolic dynamics to understand other dynamical systems. In many cases, it is a challenging problem to extract the right symbolic behavior of the dynamical system. In this example of the interval self-mapping, there is a natural symbolic framework. When proceeding from state to state , we may be moving in two possible directions: left or right. Thus, at each transition we associate the symbol L if we move left and R if we move right. A particular dynamical system then gives rise to a sequence of L's and R's. The initial state gives the sequence . Starting at x=.9 instead, we would see a sequence , since the first move would be to the left. This leads to all sorts of interesting questions about the types of sequences that arises for this dynamical system with varying c and initial state .