CH07_CHAOS

CH07_CHAOS - Chapter 7 Maps, Strange Attractors, and Chaos...

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Unformatted text preview: Chapter 7 Maps, Strange Attractors, and Chaos 7.1 Maps 7.1.1 Parametric Oscillator Consider the equation x + 2 ( t ) x = 0 , (7.1) where the oscillation frequency is a function of time. Equivalently, d dt parenleftbigg x x parenrightbigg = M ( t ) bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright parenleftbigg 1 2 ( t ) 0 parenrightbigg ( t ) bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright parenleftbigg x x parenrightbigg . (7.2) The formal solution is the path-ordered exponential, ( t ) = P exp braceleftBigg t integraldisplay dt M ( t ) bracerightBigg (0) . (7.3) Lets consider an example in which ( t ) = (1 + ) if 2 n t (2 n + 1) (1 ) if (2 n + 1) t (2 n + 2) . (7.4) Define n (2 n ). Then n +1 = exp( M ) exp( M + ) n , (7.5) 1 2 CHAPTER 7. MAPS, STRANGE ATTRACTORS, AND CHAOS Figure 7.1: Phase diagram for the parametric oscillator in the ( , ) plane. Thick black lines correspond to T = 2. Blue regions: | T | < 2. Red regions: T > 2. Magenta regions: T < 2. where M = parenleftbigg 1 2 parenrightbigg , (7.6) with (1 ) . Note that M 2 = 2 I is a multiple of the identity. Evaluating the Taylor series for the exponential, one finds exp( M t ) = parenleftBigg cos 1 sin sin cos parenrightBigg , (7.7) 7.1. MAPS 3 from which we derive Q parenleftbigg a b c d parenrightbigg = exp( M ) exp( M + ) (7.8) = parenleftBigg cos 1 sin sin cos parenrightBiggparenleftBigg cos + 1 + sin + + sin + cos + parenrightBigg with a = cos cos + + sin sin + (7.9) b = 1 + cos sin + + 1 sin cos + (7.10) c = + cos sin + sin cos + (7.11) d = cos cos + + sin sin + . (7.12) Note that det exp( M ) = 1, hence det Q = 1. Also note that P ( ) = det ( Q I ) = 2 T + , (7.13) where T = a + d = Tr Q (7.14) = ad bc = det Q . (7.15) The eigenvalues of Q are = 1 2 T 1 2 radicalbig T 2 4 . (7.16) In our case, = 1. There are two cases to consider: | T | < 2 : + = = e i , = cos 1 1 2 T (7.17) | T | > 2 : + = 1 = e , = cosh 1 1 2 | T | . (7.18) When | T | < 2, remains bounded; when | T | > 2, | | increases exponentially with time....
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This note was uploaded on 09/30/2011 for the course PHYS 221a taught by Professor Staff during the Fall '11 term at UCSD.

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CH07_CHAOS - Chapter 7 Maps, Strange Attractors, and Chaos...

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