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.

- 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_{ }.

- 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_{ }

- 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.*

See Also: Index and Initial Value Agreement

**Preferable reference for this tutorial is**

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