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Unformatted text preview: Dynamical Systems: A Brief Introduction 1. Objects of Study Phase Space M : A geometric object (e.g. sphere, tori, open set in R n ). A System T : A mapping M M . Orbits: x T ( x ) T ( T ( x )) . Ex1 : M = R 1 , T : R 1 mapsto R 1 defined by T ( x ) = 1 2 x 2 . x = 0 . 1, x 1 = 0 . 98, x 2 = . 9208, . Ex2 : M = S 1 , T ( ) = + . = 1, 1 = 1 + , 2 = 1, . Ex3 : Example: M = R 2 ,T : ( x,y ) ( x 1 ,y 1 ) T : braceleftbigg x 1 = 2 x y 1 = 1 2 y z = (1 , 1) ,z 1 = (2 , 1 2 ) ,z 2 = (2 2 , 1 2 2 ) , . Ex4 : M = R 2 , T : ( r, ) ( r 1 , 1 ) T : braceleftbigg r 1 = r 1 = + r z = (1 , 0) ,z 1 = (1 , 1) , ,z n = (1 ,n mod (2 )) , . For T : M M , and x M given, the orbit started from x is denoted as { x n } n =0 . If T 1 exists, we say that T is invertible. We then are able to talk about backward orbit from x : x 1 = T 1 x and so on. Ex1 in the above is not invertible. Ex2, Ex3 and Ex4 are invertible. 2. Fundamental Questions Q1 : Behave of individual orbits. Fixed points: T ( x ) = x . Ex: T ( x ) = x (1 x ) x = x (1 x ) x 1 = 0 , x 2 = 1 1 . Periodic orbits: T n ( x ) = x . The smallest n is the period. Ex: T ( x ) = 7 x (1 x ) Claim: Periodic orbit of all period exists. Proof: Let I 1 = [0 , 1 2 ] ,I 2 = [ 1 2 , 1]. n > 0 given, an interval I , such that (i) T i ( I ) I 1 ,i = 0 , 1 , ,n 2; (ii) T n 1 I I 2 , T n I = I 1 ....
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 Spring '10
 GLASNER

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