MAT335H1 Lecture Notes - Lecture 3: Phase Portrait, Algebraic Equation, Periodic Point

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2 Dec 2015
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The second stage of the development of chaos appeared to be at the age of the computers, and computers are most e ective when they study discrete models. Also for any practical reasons it is the di erence equations that makes practical sense. The continuous models are only used in order to facilitate the use of powerful tools of calculus; it is still debatable among physicists whether such models represent true nature of the phenomena. Now di erential dy dt = f (y) becomes di erence equation y. = f (yn) which is yn+1 yn = f (yn). yn+1 yn (n + 1) n. This is known as the di erence equation (instead of di erential equation), and is usually written as yn+1 = yn + f (yn), which in short becomes yn+1 = f (yn). These points correspond to the xed points of the function f (y), that is f (y) = y.