## Difference Equation

Dynamical system come with many different names. Our particular interesting dynamical system is for the system whose state depends on the input history. In discrete time system, we call such system difference equation (equivalent to differential equation in continuous time).

Difference equation is an equation involving differences. We can see difference equation from at least three points of views: as sequence of number, discrete dynamical system and iterated function. It is the same thing but we look at different angle.

1. Difference equation is a sequence of numbers that generated recursively using a rule to relate each number in the sequence to previous numbers in the sequence.

Example:

• Sequence is called Fibonacci sequence, generated with rule for and initial value  • Sequence has rule for . Both sequences have initial value of .
1. Difference equation is a discrete dynamical system that take some discrete input signal and produce output signal

Example:

• Dynamical system take unit step input will produce output of  1. Difference equation is an iterated map if we see the sequence as an iterated function : , The is the first iterate of under . Notation is the k-th iterate of under . For example, . The set of all iterates of is called the orbit of .

Example:

Iterated function for = 1 will produce orbit . If = 2, the iterated function generate . When = 0.5, the iterated function yield sequence of We see that knowing the rule only is not enough to know the behavior of the sequence. Initial value is also very important. The orbit of = 1 is constant for function while for = 2 produces unbounded orbit and the orbit is attracted to zero for = 0.5. The figure below show the orbit of = 0.5. We will discuss the meaning of this kind of figure later in Phase Diagram. Knowing the initial value and the rule, we can generate the whole sequence recursively. The value of is an integer ( ) and the rule to generate the sequence is called the difference equation or the dynamical system or iterated function.

Preferable reference for this tutorial is

Teknomo, Kardi (2015) Difference Equation Tutorial. https:\\people.revoledu.com\kardi\ tutorial\DifferenceEquation\