 

What is 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.
Example:
Example:
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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