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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 kth 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
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Preferable reference for this tutorial is
Teknomo, Kardi (2015) Difference Equation Tutorial. http:\\people.revoledu.com\kardi\ tutorial\DifferenceEquation\