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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 Difference Equation: What is it? is called Fibonacci sequence, generated with rule Difference Equation: What is it? for Difference Equation: What is it? and initial value Difference Equation: What is it?

Difference Equation: What is it?

  • Sequence Difference Equation: What is it? has rule Difference Equation: What is it? for Difference Equation: What is it? . Both sequences have initial value of Difference Equation: What is it? .
  1. Difference equation is a discrete dynamical system that take some discrete input signal and produce output signal


Example:

  • Dynamical system Difference Equation: What is it? take unit step input Difference Equation: What is it? will produce output of Difference Equation: What is it?

Difference Equation: What is it?

  1. Difference equation is an iterated map Difference Equation: What is it? if we see the sequence as an iterated function : Difference Equation: What is it? , Difference Equation: What is it? The Difference Equation: What is it? is the first iterate of Difference Equation: What is it? under Difference Equation: What is it? . Notation Difference Equation: What is it? is the k-th iterate of Difference Equation: What is it? under Difference Equation: What is it? . For example, Difference Equation: What is it? . The set of all iterates of Difference Equation: What is it? is called the orbit of Difference Equation: What is it? .

Example:

Iterated function Difference Equation: What is it? for Difference Equation: What is it? = 1 will produce orbit Difference Equation: What is it? . If Difference Equation: What is it? = 2, the iterated function generate Difference Equation: What is it? . When Difference Equation: What is it? = 0.5, the iterated function yield sequence of Difference Equation: What is it? 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 Difference Equation: What is it? = 1 is constant for function Difference Equation: What is it? while for Difference Equation: What is it? = 2 produces unbounded orbit and the orbit is attracted to zero for Difference Equation: What is it? = 0.5. The figure below show the orbit of Difference Equation: What is it? = 0.5. We will discuss the meaning of this kind of figure later in Phase Diagram.

Difference Equation: What is it?

Knowing the initial value and the rule, we can generate the whole sequence recursively. The value of Difference Equation: What is it? is an integer ( Difference Equation: What is it? ) 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\