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1D-dynamics - Dynamical Systems A Brief Introduction 1...

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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 = 0 . 9208, · · · . Ex2 : M = S 1 , T ( θ ) = θ + π . θ = 1, θ 1 = 1 + π , θ 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
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z = (1 , 1) , z 1 = (2 , 1 2 ) , z 2 = (2 2 , 1 2 2 ) , · · · . Ex4 : M = R 2 , T : ( r, θ ) ( r 1 , θ 1 ) T : braceleftbigg r 1 = r θ 1 = θ + r z = (1 , 0) , z 1 = (1 , 1) , · · · , z n = (1 , n mod (2 π )) , · · · . – For T : M M , and x 0 M given, the orbit started from x 0 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 0 : x 1 = T 1 x 0 and so on. – Ex1 in the above is not invertible. Ex2, Ex3 and Ex4 are invertible. 2. Fundamental Questions Q1 : Behave of individual orbits.
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– Fixed points: T ( x ) = x . Ex: T ( x ) = μx (1 x ) x = μx (1 x ) x 1 = 0 , x 2 = 1 μ 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
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(i) T i ( I ) I 1 , i = 0 , 1 , · · · , n 2; (ii) T n 1 I I 2 , T n I = I 1 . Fact: Let T : I R be continuous such that T ( I ) I (or T ( I ) I ). The T has a fixed point in I . Note that the period of the orbit constructed is n by design. Q2 : Organizations of Orbits. – Phase portrait: (orbit structure) Example A : T : braceleftbigg x 1 = 2 x y 1 = 1 2 y
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Example B : T : braceleftbigg r 1 = r θ 1 = θ + r – Local Stability: Let x 0 be a fixed point. An open neighborhood
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  • Spring '10
  • GLASNER
  • Iterated function, Dynamical systems, periodic orbit, periodic orbits, Periodic points of complex quadratic mappings

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