By Goong Chen, Yu Huang, Steven G. Krantz
This e-book contains lecture notes for a semester-long introductory graduate direction on dynamical platforms and chaos taught via the authors at Texas A&M collage and Zhongshan college, China. There are ten chapters frequently physique of the ebook, masking an common concept of chaotic maps in finite-dimensional areas. the themes comprise one-dimensional dynamical structures (interval maps), bifurcations, common topological, symbolic dynamical structures, fractals and a category of infinite-dimensional dynamical structures that are brought on by means of period maps, plus speedy fluctuations of chaotic maps as a brand new standpoint constructed by means of the authors lately. appendices also are supplied which will ease the transitions for the readership from discrete-time dynamical platforms to continuous-time dynamical structures, ruled via usual and partial differential equations. desk of Contents: easy period Maps and Their Iterations / overall diversifications of Iterates of Maps / Ordering between sessions: The Sharkovski Theorem / Bifurcation Theorems for Maps / Homoclinicity. Lyapunoff Exponents / Symbolic Dynamics, Conjugacy and Shift Invariant units / The Smale Horseshoe / Fractals / fast Fluctuations of Chaotic Maps on RN / Infinite-dimensional structures triggered by means of Continuous-Time distinction Equations
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Additional resources for Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations (Synthesis Lectures on Mathematics and Statistics)
2. It looks like a step function. The two horizontal levels correspond to the period-2 bifurcation curves in Fig. 5. Question: In the x-ranges close to x = 0 and x = 1, how oscillatory is the curve? 3. 5. It again looks like a step function, but with four horizontal levels. 16 1. 55. This curve actually has eight horizontal levels. 3. 65. This value of μ is already in the chaotic regime. The curve has exhibited highly oscillatory behavior. 18 1. 84. Note that this value of μ corresponds to the “window” area in Fig.
2n + 1 ∇ 2n + 3 ∇ .. 2·3 ∇ 2·5 ∇ 2·7 ∇ 2·9 ∇ .. ∇ 2 · (2n + 1) ∇ 2 · (2n + 3) ∇ .. 22 · 3 ∇ 22 · 5 ∇ 22 · 7 ∇ 22 · 9 ∇ .. ∇ 22 · (2n + 1) ∇ 22 · (2n + 3) ∇ .. .. 2n · 3 ∇ 2n · 5 ∇ 2n · 7 ∇ 2n · 9 ∇ .. ∇ 2n · (2m + 1) ∇ 2n · (2m + 3) ∇ .. .. .. 2m+1 ∇ 2m ∇ 2m−1 ∇ .. ∇ 23 ∇ 22 ∇ 2 ∇ 1. 1: The Sharkovski ordering. (Sharkovski’s Theorem) Let I be a bounded closed interval and f : I → I be continuous. Let n k in Sharkovski’s ordering. If f has a (prime) period n orbit, then f also has a (prime) period k orbit.
To simplify notation, let us define g(μ, y) = f (μ, y + x(μ)) − x(μ). 7) Then it is easy to check that ∂j g(μ, y) ∂y j y=0 = ∂j f (μ, x) ∂x j x=x(μ) for , j = 1, 2, 3, . . 8) This change of variable will give us plenty of convenience. We note that y = y(μ) ≡ 0 becomes the curve of fixed points for the map g. Since g(μ, y)|y=0 = g(μ, 0) = 0, we have the Taylor expansion g(μ, y) = a1 (μ)y + a2 (μ)y 2 + a3 (μ)y 3 + O(|y|4 ). The period-2 points of fμ now satisfy y = gμ2 (y) = g(μ, g(μ, y)) = a1 (μ)[a1 (μ)y + a2 (μ)y 2 + a3 (μ)y 3 + O(|y|4 )] + a2 (μ)[a1 (μ)y + a2 (μ)y 2 + a3 (μ)y 3 + O(|y|4 )]2 + a3 (μ)[a1 (μ)y + a2 (μ)y 2 + a3 (μ)y 3 + O(|y|4 )]3 + O(|y|4 ) y = a12 y + (a1 a2 + a12 a2 )y 2 + (a1 a3 + 2a1 a22 + a13 a3 )y 3 + O(|y|4 ), where in the above, we have omitted the dependence of a1 , a2 and a3 on μ.