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

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Chapter 7 Maps, Strange Attractors, and Chaos 7.1 Maps 7.1.1 Parametric Oscillator Consider the equation ¨ x + ω 2 0 ( 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 0 1 ω 2 0 ( t ) 0 parenrightbigg ϕ ( t ) bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright parenleftbigg x ˙ x parenrightbigg . (7.2) The formal solution is the path-ordered exponential, ϕ ( t ) = P exp braceleftBigg t integraldisplay 0 dt M ( t ) bracerightBigg ϕ (0) . (7.3) Let’s consider an example in which ω ( t ) = (1 + ǫ ) ω 0 if 2 t (2 n + 1) τ (1 ǫ ) ω 0 if (2 n + 1) τ t (2 n + 2) τ . (7.4) Define ϕ n ϕ (2 ). Then ϕ n +1 = exp( M τ ) exp( M + τ ) ϕ n , (7.5) 1
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2 CHAPTER 7. MAPS, STRANGE ATTRACTORS, AND CHAOS Figure 7.1: Phase diagram for the parametric oscillator in the ( θ, ǫ ) plane. Thick black lines correspond to T = ± 2. Blue regions: | T | < 2. Red regions: T > 2. Magenta regions: T < 2. where M ± = parenleftbigg 0 1 ω 2 ± 0 parenrightbigg , (7.6) with ω ± (1 ± ǫ ) ω 0 . Note that M 2 ± = ω 2 ± · I is a multiple of the identity. Evaluating the Taylor series for the exponential, one finds exp( M ± t ) = parenleftBigg cos ω ± τ ω 1 ± sin ω ± τ ω ± sin ω ± τ cos ω ± τ parenrightBigg , (7.7)
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7.1. MAPS 3 from which we derive Q ≡ parenleftbigg a b c d parenrightbigg = exp( M τ ) exp( M + τ ) (7.8) = parenleftBigg cos ω τ ω 1 sin ω τ ω sin ω τ cos ω τ parenrightBiggparenleftBigg cos ω + τ ω 1 + sin ω + τ ω + sin ω + τ cos ω + τ parenrightBigg with a = cos ω τ cos ω + τ ω + ω sin ω τ sin ω + τ (7.9) b = 1 ω + cos ω τ sin ω + τ + 1 ω sin ω τ cos ω + τ (7.10) c = ω + cos ω τ sin ω + τ ω sin ω τ cos ω + τ (7.11) d = cos ω τ cos ω + τ ω ω + sin ω τ sin ω + τ . (7.12) Note that det exp( M ± τ ) = 1, hence det Q = 1. Also note that P ( λ ) = det ( Q − λ · I ) = λ 2 + Δ , (7.13) where T = a + d = Tr Q (7.14) Δ = ad bc = det Q . (7.15) The eigenvalues of Q are λ ± = 1 2 T ± 1 2 radicalbig T 2 4 Δ . (7.16) In our case, Δ = 1. There are two cases to consider: | T | < 2 : λ + = λ = e , δ = cos 1 1 2 T (7.17) | T | > 2 : λ + = λ 1 = ± e μ , μ = cosh 1 1 2 | T | . (7.18) When | T | < 2, ϕ remains bounded; when | T | > 2, | ϕ | increases exponentially with time. Note that phase space volumes are preserved by the dynamics. To investigate more fully, let θ ω 0 τ . The period of the ω 0 oscillations is δt = 2 τ , i.e. ω pump = π/τ is the frequency at which the system is ‘pumped’. We compute the trace of Q and find 1 2 T = cos(2 θ ) ǫ 2 cos(2 ǫθ ) 1 ǫ 2 . (7.19) We are interested in the boundaries in the ( θ, ǫ ) plane where | T | = 2. Setting T = +2, we write θ = + δ , which means ω 0 pump . Expanding for small δ and ǫ , we obtain the relation δ 2 = ǫ 4 θ 2 ǫ = vextendsingle vextendsingle vextendsingle vextendsingle δ vextendsingle vextendsingle vextendsingle vextendsingle 1 / 2 . (7.20)
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4 CHAPTER 7. MAPS, STRANGE ATTRACTORS, AND CHAOS Setting
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