PHY411 Lecture notes Part 6 Alice Quillen September 17, 2015 Contents 0.1 Introduction . 1 1 Circle maps and Arnold Tongues 2 1.1 Sine-circle map . 4 1.2 K < 1 quasi-periodic and mode-locking behavior . 8 1.3 K = 1 Critical value giving a Cantor set . 9 1.4 K > 1 Period doubling and chaos . 9 1.5 Perturbations . 10 1.6 Density maps . 10 2 Ways to describe rational and irrational numbers 11 2.1 Farey sequence . 11 2.2 Continued Fractions . 12 2.3 Irrational Numbers and Diophantine approximation . 12 2.4 Pigeonholes . 13 3 Newton's method 17 3.1 Quadratic functions . 18 3.2 Convergence of Newton's map . 20 3.3 Newton's method as a first order approximation . 21 0.1 Introduction The material here is taken from a variety of sources. Newton's map section following Holmgren's book. Circle map stuff from some reviews (H. Bruin, Physica D, 1992, and a review by Predrag Cvitanovic). The list of rational/irrational number discussion is currently like a laundry list and not yet nicely related to the dynamical problems. It would help to add something on the circle map done with perturbation theory (as by Rajeev) to illustrate some KAM-like ideas and connect the rational/irrational number discussion 1 to the dynamics. I had not yet added any complex number dynamics but Milnor's book on the topic is nicely accessible. The set of topics here is nicely set up so that I can in future add a section on KAM theory for maps of the circle (following original paper by Arnold) that includes a holonomic type of equation (that will look familiar in the context of near-integrable Hamiltonian KAM theory). I another set of notes I looked a some nice invertible maps of the circle in an attempt to try to converge to a topological conjugacy without doing Fourier expansion. The exploration of the circle map is interesting because in some regimes, the map is either like a rotation or has periodic orbits depending upon whether its winding number is rational or irrational. The circle map provides an analogy to weakly perturbed nearly integrable Hamiltonian systems. A perturbed Hamiltonian system can be transformed via a series of canonical transformations to resemble an integrable system if the unperturbed system is sufficiently far away from resonances. This is the focus of Kolmogorov's theorem and part of a body of work known as KAM theory. The resonances in the perturbed Hamiltonian setting are played by the role of rational winding numbers for the circle map. Instead of using canonical transformations to transform the Hamiltonian system, perturbation techniques on the circle map can be used to find a topological conjugacy to a rotation map. A procedure giving the map (giving the topological conjugacy) will only converge if the winding number is sufficiently irrational. Likewise the procedure giving the canonical transformation in the Hamiltonian setting will only converge if the frequencies are distant from resonances. 1 Circle maps and Arnold Tongues A dissipative oscillator can have asymptotic motion that is periodic. The attractor in this case is called a limit cycle. When two oscillators are coupled together, the motion is in a 2-d space equivalent to a torus. When the oscillators mode-lock, the attractor of the combined system is a periodic orbit on the torus, and when they unlock the trajectory is quasi periodic, densely covering the torus. Taking a Poincar´esection reduces the torus to a circle. The flow on the torus in this surface of section becomes an iterated mapping of the circle with a single iteration of the resulting circle map corresponding to a complete revolution about one axis of the torus. A map from the unit interval back onto itself can be regarded as a map of the circle onto itself. The map θn+1 = [θn + Ω] mod 1 (1) corresponds to a shift or rotation by Ω. This is in the class of maps θn+1 = f(θn) where f(θ) is a periodic function on the unit interval f(θ + 1) = f(θ). 2 We can define a winding number W (sometimes called the rotation number) as N−1 1 X W ≡ lim ∆θn (2) n!1 N n=0 with ∆θn = [θn+1 − θn] mod 1 For the rotation map in equation 1 the winding number should be equal to Ω. Sometimes the winding number is written as 1 W = lim θn (3) n!1 n The two definitions are should be equivalent, but care should be taken with the modulo operation. For example for W computed with Equation 2, if the differences ∆θn vary alternate between and 1 − the winding number computed would converge to 0.5 but it should be 0 or 1. The easiest way to compute the winding number sensibly is to use a lift to the real line. In other words let φn+1 = f(φn) with no modulo operator. We no longer require f to be periodic. This sends numbers on the real line to numbers on the real line. The circle map is recovered with the modulo operation θn = φn mod 1 sending intervals on the real line into the unit interval. For a winding number W , φn would on average keep increasing by W with each application of the map. So we can compute the winding number with 1 W = lim φn N!1 n and by not taking the modulo during the mapping. So if you see the winding number written as given by equation 3 it is actually computed without the modulo operation. If the winding number is rational and equal to W = p=q then there are periodic fixed points with period q. The winding number depends on a limit so if the winding number is rational then orbit must converge to a periodic orbit. If the winding number is irrational, then there are no periodic orbits. The behavior is called quasi-periodic. These orbits are dense. This means that given any point and any size neighborhood containing that point, the orbit will eventually come into that neighborhood. 3 1.1 Sine-circle map We now look at maps that are more complicated than a rotation or a shift θn+1 = [θn + Ω + g(θn)] mod 1 where g(0) = g(1) so g is a periodic function g(θ + 1) = g(θ). We can also assert that R 1 0 g(θ)dθ = 0 as the constant or average value for g can be incorporated in the rotation Ω. It is useful to consider the lifted map on the real line φn+1 = f(φn) with f(φ) = φ + Ω + g(φ) As the integral of g is zero, repeated operations of the lifted map with Ω > 0 should cause φ to increase forever, giving a non-zero winding number, computed as 1 W = lim f(φn) n!1 n An example of a circle map, is the much studied sine-map giving rise to a phenomena called Arnold tongues K θ = [θ + Ω − sin(2πθ )] mod 1 n+1 n 2π n This map we can write in terms of the function K f (θ) = θ + Ω − sin(2πθ) Ω;K 2π If K ≤ 1 the map is invertible. This limit for being invertible can be understood by looking at the fixed point with f 0(θ) = 0. When K is near zero we recover our shift or rotation map. When K is not zero (but not greater than 1) for each rational number there is a region in θ where the winding number is equal to that rational number. The region is mode locked. The recurrence time N for a particular is the smallest integer N such that kθN − θ0k < and is a good way to numerically identify the rational winding numbers W = p=q with low q (see Figure 2). Here kθN − θ0k should be computed so that it is a minimum of jθN − θ0j; jθN − θ0 + 1j; jθN − θ0 − 1j or a minimum of (θN − θ0) mod 1. 4 Figure 1: a) The winding number with black corresponding to W = 0 and red W = 1 with green W = 1=2. x axis is Ω and ranges from 0-1. y-axis is K and ranges from 0 to 4π. Original artwork created by Linas Vepstas (from wiki). b) Winding or rotation number as a function of Ω for K = 1, also known as the devil's staircase. I computed this with K = 1 − 10−3 and using 300 iterations of the map. 5 Figure 2: a) The recurrence time N is plotted for = 0:003 with black N < 10, blue values of about 50, green shows values of about 140 and red for N > 250. x axis is Ω and ranges from 0-1. y-axis is K and ranges from 0 to 4π. b) Bifurcation map for Ω = 1=3. Here the x axis is θn plotted for many iterations. Again the y axis is K 2 (0; 4π). Original artwork created by Linas Vepstas (from wiki). 6 Figure 3: My attempt to compute the winding number using equation 3. After doing this I was much more impressed with Linas Vepstas's artwork! 7 1.2 K < 1 quasi-periodic and mode-locking behavior We can worry about convergence of the winding number. Let N−1 X ∆θn W ≡ N n n=0 If the series lim WN N!1 converges then the winding number exists and is either rational or irrational and behavior is either quasi-periodic or mode-locked, respectively.