By Shlomo Sternberg
A pioneer within the box of dynamical platforms created this contemporary one-semester creation to the topic for his periods at Harvard collage. Its wide-ranging remedy covers one-dimensional dynamics, differential equations, random walks, iterated functionality platforms, symbolic dynamics, and Markov chains. Supplementary fabrics provide numerous on-line parts, together with PowerPoint lecture slides and MATLAB routines. 2010 version.
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Extra resources for Dynamical Systems (June 4, 2009 Draft)
Before embarking on the study of bifurcations let us observe that if p is a fixed point of Fµ and Fµ (p) = 1, then for ν close to µ, the transformation Fν has a unique fixed point close to p. Indeed, the implicit function theorem applies to the function P (x, ν) := F (x, ν) − x since ∂P (p, µ) = 0 ∂x by hypothesis. We conclude that there is a curve of fixed points x(ν) with x(µ) = p. The first type of bifurcation we study is the fold bifurcation where there is no (local) fixed point on one side of the bifurcation value, b, where a fixed point p appears at µ = b with Fµ (p) = 1, and at the other side of b the map Fµ has two fixed points, one attracting and the other repelling.
To complete the proof we must show that this critical point is a local maximum. So we must compute the second derivative at the origin. Step VI, completion of the proof. Calling this function φ we have φ(x) := ∂F ◦2 (x, ν(x)) ∂x φ (x) = ∂ 2 F ◦2 ∂ 2 F ◦2 (x, ν(x)) + (x, ν(x))ν (x) 2 ∂x ∂x∂µ φ (x) = ∂ 3 F ◦2 ∂ 3 F ◦2 (x, ν(x)) + 2 (x, ν(x)ν (x) ∂x3 ∂x2 ∂µ ∂ 2 F ◦2 ∂ 3 F ◦2 2 + (x, ν(x))(ν (x)) + (x, ν(x))ν (x). ∂x∂µ2 ∂x∂µ The middle two terms vanish at 0 since ν (0) = 0. The last term becomes dλ (0)ν (0) < 0 dµ by condition (e) and the fact that ν (0) < 0.
See, for example, [McMullen]. Chapter 3 Sarkovsky’s theorem, Singer’s theorem, intermittency. √ The logistic map Lµ develops a period three orbit as µ increases above 1 + 8, and this orbit is initially stable. According to a theorem of Sarkovsky, if a continuous map of a compact interval on the real line has an orbit of period three, then it has orbits √ of all periods. Nevertheless, we do not see these other periodic orbits near 1 + 8. The reason is that they are all unstable. In fact, a theorem of Singer implies that Lµ can have at most one stable periodic orbit.
Dynamical Systems (June 4, 2009 Draft) by Shlomo Sternberg