Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations by Goong Chen, Yu Huang, Steven G. Krantz

By Goong Chen, Yu Huang, Steven G. Krantz

This ebook comprises lecture notes for a semester-long introductory graduate direction on dynamical platforms and chaos taught by way of the authors at Texas A&M collage and Zhongshan collage, China. There are ten chapters basically physique of the ebook, masking an straightforward thought of chaotic maps in finite-dimensional areas. the subjects contain one-dimensional dynamical platforms (interval maps), bifurcations, basic topological, symbolic dynamical structures, fractals and a category of infinite-dimensional dynamical platforms that are brought about by way of period maps, plus speedy fluctuations of chaotic maps as a brand new perspective constructed through the authors in recent times. appendices also are supplied with a view to ease the transitions for the readership from discrete-time dynamical platforms to continuous-time dynamical structures, ruled by means of traditional and partial differential equations. desk of Contents: basic 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 platforms prompted by means of Continuous-Time distinction Equations

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Where k ≡ k0 · min{|I1 |, |I2 |} and √ 1+ 5 α ≡ ln 2 > 0. 5, we can also establish the following. Let I be a bounded closed interval and f : I → I be continuous. Assume that I1 , I2 , . . 6 I1 −→ I2 −→ I3 −→ · · · −→ In −→ I1 ∪ Ij , for some j = 1. 13) Then for some K > 0 and α > 0, VI (f n ) ≥ Keαn −→ ∞, as n → ∞. 6 motivates us to give the following definition of chaos, the first one of such definitions in this book. 14) holds. We say that f is chaotic in the sense of exponential growth of total variations of iterates.

Also, 0= d2 ∂ 2f ∂ 2f ∂f ∂ 2f 2 m m G(m(x), x) = [m (x)] + 2 (x) + (x) + = 0. 22). Finally, we analyze stability of points on C near (μ, x) = (μ0 , x0 ). The stability is determined by whether ∂ f (μ, x) is less than 1 or greater than 1. ∂x μ=m(x) x We have ∂ f (μ, x) ∂x μ=m(x) ∂ ∂ 2 f (μ0 , x0 ) (x − x0 ) f (μ0 , x0 ) + ∂x ∂x 2 ∂ 2 f (μ0 , x0 ) + (m(x) − μ0 ) ∂μ∂x 3 1 ∂ f (μ0 , x0 ) + (x − x0 )2 2! ∂x 3 ∂ 3 f (μ0 , x0 ) + (m(x) − μ0 )(x − x0 ) ∂μ∂x 2 1 ∂ 3 f (μ0 , x0 ) (m(x) − μ0 )2 + · · · + 2!

5, we can also establish the following. Let I be a bounded closed interval and f : I → I be continuous. Assume that I1 , I2 , . . 6 I1 −→ I2 −→ I3 −→ · · · −→ In −→ I1 ∪ Ij , for some j = 1. 13) Then for some K > 0 and α > 0, VI (f n ) ≥ Keαn −→ ∞, as n → ∞. 6 motivates us to give the following definition of chaos, the first one of such definitions in this book. 14) holds. We say that f is chaotic in the sense of exponential growth of total variations of iterates. 7 In Chapter 9, such a map f will also be said to have rapid fluctuations of dimension 1.

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