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What is Difference Equation?

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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.

  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.

 

See Also: Index and Initial Value Agreement

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