By Shlomo Sternberg
A pioneer within the box of dynamical structures created this contemporary one-semester advent to the topic for his sessions at Harvard collage. Its wide-ranging therapy covers one-dimensional dynamics, differential equations, random walks, iterated functionality structures, symbolic dynamics, and Markov chains. Supplementary fabrics supply various on-line elements, together with PowerPoint lecture slides and MATLAB routines. 2010 version.
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Additional info for Dynamical Systems (June 4, 2009 Draft)
5. For any value of µ the fixed points of Lµ are 0 and 1 − µ1 . Since Lµ (x) = µ − 2µx, 1 Lµ (0) = µ, Lµ (1 − ) = 2 − µ. 1 0 < µ ≤ 1. For 0 < µ < 1, 0 is the only fixed point of Lµ on [0, 1] since the other fixed point, 1 − µ1 , is negative. On this range of µ, the point 0 is an attracting fixed point since 0 < Lµ (0) = µ < 1. Under iteration, all points of [0, 1] tend to 0 under the iteration. The population “dies out”. 2 µ = 1. For µ = 1 we have L1 (x) = x(1 − x) < x, ∀x > 0. Each successive application of L1 to an x ∈ (0, 1] decreases its value.
Hence it maps the interval [1 − µ1 , µ4 ] into an interval whose right hand end point is 1 − µ1 and whose left hand end point is Lµ ( µ4 ). We claim that µ 1 Lµ ( ) > . 4 2 This amounts to showing that µ2 (4 − µ) 1 > 16 2 or that µ2 (4 − µ) > 8. So we need only check the values of µ2 (4 − µ) at the end points, 2 and 3, of the range of µ we are considering, where the values are 8 and 9. So we have proved that the image of [ µ1 , 1 − µ1 ] is the same as the image of [ 12 , 1− µ1 ] and is [1− µ1 , µ4 ].
5 and is superattractive. As µ increases, the fixed point continues to move to the right. 5. 23606797... 5 is a period two point, and so the period two points √ are superattractive. 449.. the period two points have become repelling and attracting period four points appear. In fact, this scenario continues. The period 2n−1 points appear at bifurcation values bn . They are initially attracting, and become superattracting at sn > bn and become unstable past the next bifurcation value bn+1 > sn when the period 2n points appear.
Dynamical Systems (June 4, 2009 Draft) by Shlomo Sternberg