By Goong Chen, Yu Huang, Steven G. Krantz
This publication includes lecture notes for a semester-long introductory graduate direction on dynamical structures and chaos taught by way of the authors at Texas A&M college and Zhongshan college, China. There are ten chapters primarily physique of the publication, protecting an straightforward concept of chaotic maps in finite-dimensional areas. the subjects comprise one-dimensional dynamical structures (interval maps), bifurcations, basic topological, symbolic dynamical platforms, fractals and a category of infinite-dimensional dynamical structures that are prompted via period maps, plus swift fluctuations of chaotic maps as a brand new point of view built via the authors in recent times. appendices also are supplied with a view to ease the transitions for the readership from discrete-time dynamical structures to continuous-time dynamical structures, ruled via traditional and partial differential equations. desk of Contents: basic period Maps and Their Iterations / overall diversifications of Iterates of Maps / Ordering between classes: 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 caused by way of Continuous-Time distinction Equations
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Additional resources for Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations
1), where I2j −1 = [x2j −2 , x2j ] and I2j = [x2j +1 , x2j −1 ], for j = 2, . . , (n − 1)/2. We now look at the covering relations of intervals I1 , I2 , . . , In . From f (x1 ) = x2 , f (x2 ) = x3 , we have f (I1 ) ⊇ I1 ∪ I2 . 3) From f (x3 ) = x4 , f (x1 ) = x2 , we have f (I2 ) ⊇ I3 . 4) f (I3 ) ⊇ I4 , f (I4 ) ⊇ I5 , . . , f (In−2 ) ⊇ In−1 . 5) Similarly, 31 However, In−1 = [xn , xn−2 ], f (xn ) = x1 , f (xn−2 ) = xn−1 , and, therefore f (In−1 ) ⊇ [x1 , xn−1 ] = I1 ∪ I3 ∪ I5 ∪ · · · ∪ In−2 .
For example, Newton’s method for finding roots of a nonlinear equation, and the time-marching of a 1-step explicit Euler finitedifference scheme for a first order scalar ordinary differential equation, can both result in an interval map. Interestingly, even for partial differential equations such as the nonlinear initial-boundary value problem of the wave equation in Appendix B, interval maps have found good applications. 4) as a standard example to illustrate many peculiar, amazing behaviors of the iterates of the quadratic map.
This function describes a curve C of fixed points near (μ0 , x0 ). We have 0= ∂f ∂f d G(m(x), x) = m (x) + − 1 = 0. dx ∂μ ∂x At (μ, x) = (μ0 , x0 ), 0= ∂f (μ0 , x0 ) ∂f (μ0 , x0 ) ∂f (μ0 , x0 ) m (x0 ) + −1= m (x0 ), ∂μ ∂x ∂μ and so m (x0 ) = 0. 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!
Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations by Goong Chen, Yu Huang, Steven G. Krantz