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Lecture 18 - Introduction to Complex Dynamics - 1/3 Math 207 - Spring '17 - Fran¸coisMonard 1 Lecture 18 - Introduction to complex dynamics - 1/3 Outline: • Definition of a discrete dynamical system F : C^ ! C^ (or R ! R), and the typical questions of interest (long-term behavior of orbits, bifurcations ?) Examples of 1-dimensional systems. • Applications: finding roots of polynomials by Newton's method: iteration of rational functions on C^ ! • Examples: f(z) = z + 1; f(z) = λz; f(z) = zd; f pol. degree d. • Any convergent orbit converges to a fixed point of F . • Definition: fixed points, k-periodic points (∼ fixed points of F k). • Multiplier and stability: attracting, superattracting, repelling or neutral. • Examples: f(x) = x2, f(x) = x2 − 1. • If a fixed point z0 is attracting or superattracting, there exists a small ball where every seed n converges to the same point. (fF g converges normally to z0). Repelling ? Neutral ? ^ n n!1 • Basin of attraction AF (z0) := fz 2 C;F (z) −! z0g. k 0 k 0 • Multiplier for a k-cycle: λ = (F ) (z0) = ··· = (F ) (zk−1). • One-dimensional dynamics, graphical analysis. Graphical classification of fixed points. Study of the quadratic family for c ≥ −2. ^ ^ 1 ^ Given F : C ! C and z0 2 C (the seed), F generates a discrete dynamical system, i.e. a n sequence-generating mechanism via the rule zn+1 = F (zn). The sequence fzn = F (z0)g is called n the orbit of z0 under F . Here we denote by F the n-fold composition F ◦ · · · ◦ F and this should n not be confused with the complex number (F (z0)) . n plays the role of a discrete time variable. Example 1. Iterating a function appears naturally in Newton's method, whose purpose is to find solutions of an equation f(x) = 0 for f given, by setting up the iterative scheme xn+1 = xn − f(xn) 0 . Since we have no explicit method a priori to find the zeros of a general function (except for f (xn) polynomials of order up to 5), the recursion relation above means that, starting at a seed x0, we pick x1 as the unique zero of the linear function approximating f at x0 (i.e. the Taylor expansion 0 p(x) = f(x0) + (x − x0)f (x0)), then we repeat this procedure at x1 and so on. In this case, the f(x) map that is being iterated over is F (x) := x − f 0(x) , called the Newton map of f. The study of Newton's algorithm was one of the main motivations for studying iteration of functions [A]. Typical questions that one may ask are: 1. What is the long-term behavior of the orbits of a given dynamical system zn+1 = F (zn)? Possible answers are: convergent, divergent, periodic, asymptotically periodic, none of the above (leading to rather erratic trajectories) 1 Here C^ can be replaced by subsets of it, or other topological spaces. Math 207 - Spring '17 - Fran¸coisMonard 2 2. Given a one-parameter family of dynamical systems zn+1 = Fc(zn), how does the answer to question 1 vary in terms of the values of c ? This is the question behind bifurcation theory. 2 In particular, we will spend some time studying the family Fc(x) = x + c for c 2 R and x 2 R. We will then complexify both z and c. General concepts Fixed points. We start with the following observation. Here and below, we fix F a continuously differentiable function F . n ? ? Theorem 1. If an orbit zn = F (x0) converges to z 2 C, then z is a fixed point of F , i.e. F (z?) = z?. ? ? Proof. F (z ) = F (limn!1 zn) = limn!1 F (zn) = limn!1 zn+1 = z , where we have used the continuity of F at z?. Hence we see that a dynamical system has convergent orbits only if it has fixed points, this also ? gives us an obvious (constant) orbit fzn = z gn for this dynamical system. Other \closed orbits" consists of k-cycles made up of k-periodic points: k j Definition 1. z0 is a k-periodic point of F if F (z0) = z0 and for every j < k, F (z0) 6= z0. The orbit of z0 is a k-cycle (z0; z1; : : : ; zk−1; z0; z1;::: ), each point of which is a k-periodic point of F . Stability. The next notion then concerns stability of these constant orbits: what happens if we ? ? perturb the seed z0 a little bit away from a fixed point z , does the orbit still converge to z ? Definition 2. A fixed point of F is called attracting if jF 0(z?)j < 1; superattracting if F 0(z?) = 0; repelling if jF 0(z?)j > 1; neutral if jF 0(z?)j = 1. The number λ = F 0(z?) is also referred to as the multiplier of z?. We can also define a multiplier for a k-cycle, and this goes as follows: Theorem 2. If (z0; : : : ; zk−1) is a k-cycle of F , we have k 0 k 0 k 0 (F ) (z0) = (F ) (z1) = ··· = (F ) (zk−1): Proof. Chain rule. The number above is then called the multiplier of this cycle, and one may define the same attributes of this cycle (attracting, etc. ) depending on the values of that multiplier. The next theorem justifies the terminology \attracting" above. ? ? Theorem 3. If z is attracting, then there exists ρ > 0 such that for every z0 2 Dρ(z ), the orbit ? of z0 converges to z . A similar conclusion holds for the real-valued case. You may think about expressing similar conclusions for the case of attracting k-cycles. Math 207 - Spring '17 - Fran¸coisMonard 3 Proof. By continuity of F 0, since jF 0(z?)j < 1, then jF 0(z)j ≤ c < 1 in a small neighbourhood ? ? ? Dρ(z ) of z . Then for any z 2 Dρ(z ), jF (z) − F (z?)j ≤ Cjz − z?j; i.e. F is a contraction there, and by the contraction mapping theorem, any seed has its orbit converge to z?. Remark 1. Similarly to this proof, we can understand the concept of repelling, in which case one would have instead jF (z) − F (z?)j > βjz − z?j near z? for some β > 1, thereby preventing any sequence starting near z? to converge to z?. This motivates the notion of basin of attraction: ? ? Definition 3. For z a fixed point of F : C^ ! C^, the basin of attraction of z is defined by ? n ? AF (z ) = fz 2 ^ : lim F (z) = z g: C n!1 Remark 2. If z? is superattracting, then jF (z)−F (z?)j ≈ βjz −z?jp for z near z? and some p ≥ 2, so that convergence to z? becomes much faster than linear. The conclusion is not as straightforward when a fixed point is neutral, as we will see below. Real dynamics and graphical analysis Let's draw some lessons from iterating real-valued maps, and consider a dynamical system F : R ! R. We now introduce a brief method to easily draw trajectories. The idea is to plot the function F together with the line fy = xg. Drawing the orbit of a given x0 can be done by following the strategy below: start from x0 on the real axis, move vertically toward the graph of f, then move horizontally toward the line fy = xg and repeat the process. The sequence xn can be read as all the successive abscissae of the points hit on the line fy = xg. See typical situations on Fig.1. Figure 1: Examples of graphical analyses in the case of an attracting point (left) with F (x) = :05x2 + :25x + 1 and a repelling point (right) for F (x) = 2x − :03x2. Remark on neutral fixed points: Using this tool, it is easy to realize that neutral fixed points can exhibit all possible behaviors: pick a function F whose graph is tangent to fy = xg at some x?. Math 207 - Spring '17 - Fran¸coisMonard 4 • To the left of x?, if the graph of F lies above the curve fy = xg, graphical analysis will show that nearby seeds will converge to x?. If the graph of F lies below the curve fy = xg, nearby seeds will diverge away from x?. • To the right of x?, the opposite situation happens. For each situation above, since there are functions whose graphs can achieve either situation, the overall behavior of a neutral point can be: attracting, repelling, or attracting on one side and repelling on the other. Example 2. Let F (x) = x2. Find all fixed points of F and classify them. By graphical analysis, deduce the long-term behavior of all orbits. Solution: F (x) = x is equivalent to x(x − 1) = 0 so the fixed points are 0 and 1. F 0(0) = 0 so x? = 0 is superattracting and F 0(1) = 2 so x? = 1 is repelling. Graphical analysis arrives at the following conclusions: • if x 2 (−1; 1), the orbit of x converges to 0. • if x 2 {±1g, the orbit of x converges to 1. n • if jxj > 1, then limn!1 F (x) = +1. In this case, we have AF (0) = (−1; 1), AF (1) = {±1g, which tells us that basins of attraction can be topologically quite diverse (AF (0) is open and not closed, Af (1) is closed and not open). Moreover, if we extended F as a meromorphic function on the extended complex plane, it would make sense to say that 1 is another fixed point of F (F (1) = 1) so we could talk of the basin of attraction of 1 and say that AF (1) = (−∞; −1) [ (1; +1).
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