Math 207 - Spring ’17 - Fran¸coisMonard 1

Lecture 18 - Introduction to complex dynamics - 1/3

Outline:

• Definition of a discrete F : Cˆ → Cˆ (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 Cˆ ! • Examples: f(z) = z + 1; f(z) = λz; f(z) = zd; f pol. degree d.

• Any convergent 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. ({F } converges normally to z0). Repelling ? Neutral ? ˆ n n→∞ • Basin of attraction AF (z0) := {z ∈ C,F (z) −→ z0}. 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.

ˆ ˆ 1 ˆ Given F : C → C and z0 ∈ 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 {zn = F (z0)} 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 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 Cˆ 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 . 2 In particular, we will spend some time studying the family Fc(x) = x + c for c ∈ R and x ∈ 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 ∈ C, then z is a fixed point of F , i.e. F (z?) = z?.

? ? Proof. F (z ) = F (limn→∞ zn) = limn→∞ F (zn) = limn→∞ 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 {zn = z }n 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 |F 0(z?)| < 1; superattracting if F 0(z?) = 0; repelling if |F 0(z?)| > 1; neutral if |F 0(z?)| = 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 ∈ 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 |F 0(z?)| < 1, then |F 0(z)| ≤ c < 1 in a small neighbourhood ? ? ? Dρ(z ) of z . Then for any z ∈ Dρ(z ),

|F (z) − F (z?)| ≤ C|z − z?|, 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 |F (z) − F (z?)| > β|z − z?| 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 : Cˆ → Cˆ, the basin of attraction of z is defined by

? n ? AF (z ) = {z ∈ ˆ : lim F (z) = z }. C n→∞ Remark 2. If z? is superattracting, then |F (z)−F (z?)| ≈ β|z −z?|p 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 {y = x}. 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 {y = x} and repeat the process. The sequence xn can be read as all the successive abscissae of the points hit on the line {y = x}. 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 {y = x} at some x?. Math 207 - Spring ’17 - Fran¸coisMonard 4

• To the left of x?, if the graph of F lies above the curve {y = x}, graphical analysis will show that nearby seeds will converge to x?. If the graph of F lies below the curve {y = x}, 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 ∈ (−1, 1), the orbit of x converges to 0.

• if x ∈ {±1}, the orbit of x converges to 1.

n • if |x| > 1, then limn→∞ F (x) = +∞.

In this case, we have AF (0) = (−1, 1), AF (1) = {±1}, 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 ∞ is another fixed point of F (F (∞) = ∞) so we could talk of the basin of attraction of ∞ and say that AF (∞) = (−∞, −1) ∪ (1, +∞). Example 3. Same instructions as above for the function F (x) = x2 − 1. √ Solution: Searching the fixed points of F yields x = x := 1± 5 . Then F 0(x ) = 2x = √ √ ± 2 + + 0 1 + 5 > 1 and F (x−) = 1 − 5 < −1 so both fixed points are repelling. By graphical analysis, we n n see that if |x| > x+, we again have limn→∞ F (x) = +∞, if x = ±x+, then limn→∞ F (x) = x+, n and if x = ±x−, limn→∞ F (x) = x−. On the other hand all other points seems to generate an orbit which eventually alternates between the values 0 and −1. Let us classify the 2-cycle (0, −1). F 2(x) = F (F (x)) = (x2 − 1)2 − 1 = x4 − 2x2. (F 2)0(x) = 4x(x2 − 1) is such that (F 2)0(0) = (F 2)0(−1) = 0, so the cycle is indeed superattracting and attracts all seeds in the set (−x+, x+) except at ±x−. Remark 3. We may notice that F 2 above is a polynomial of degree 4, so F (x) − x = 0 has two other roots, i.e. F 2 has two other fixed points. Could we have forgotten another 2-cycle along the way ? No, the two remaining points are the first two fixed points of F itself x+ and x− (if F (x) = x, then certainly F 2(x) = F (F (x)) = F (x) = x), which can be seen as two 1-cycles.

The quadratic family

Examples2 and3 seem to come from very similar functions ( x2 and x2 − 1), yet the dynamical behaviors exhibited show quite different features. Considering now the one-parameter family of Math 207 - Spring ’17 - Fran¸coisMonard 5

2 functions Fc(x) = x + c for c ∈ R, we will see that varying continuously the parameter c can lead to drasting changes in dynamical behavior as c passes certain values.

As in both examples considered above, let us start by searching the fixed points of Fc. Fc(x) = x 1 2 1 1 is equivalent to (x − 2 ) + c − 4 = 0. A first observation, then, is that if c > 4 , Fc has no fixed point. Graphically, the parabola lies completely above the curve {y = x}, and graphical analysis shows that all orbits have limit +∞. The natural way of evolving c is then to dial it down starting 1 from above 4 and see what happens to the dynamics.

1 The saddle-node bifurcation through c = 1/4. When c = 4 , the parabola is tangent to 1 1 ? 1 0 1 the curve {y = x} at the point ( 2 , 2 ), and x = 2 is a fixed point with F 1 ( 2 ) = 1, so the point is 4 neutral. By graphical analysis, we arrive at the following conclusion: 1 • if |x| > 2 , the orbit of x has limit +∞. 1 1 • if |x| ≤ 2 , the orbit of x has limit 2 . √ 1 1 q 1 0 When c < , Fc has now two fixed points x± = ± − c with F (x+) = 1 + 1 − 4c > 1 4 2 4 √ c 1 0 so x+ is repelling for all c < 4 . On the other hand, Fc(x−) = 1 − 1 − 4c, so x− can still exhibit 0 all types of stability depending on the value of c. It is always true that Fc(x−) < 1 but forcing 0 3 −3 1 Fc(x−) > −1 entails c > 4 . This means that in the regime 4 < c < 4 , x− is either attracting or superattracting. Graphical analysis now yields the following conclusions: • If |x| > x+, the orbit of x has limit +∞. • If x = ±x+, the orbit of x has limit x+. • If |x| ≤ x−, the orbit of x has limit x−.

Figure 2: The saddle-node bifurcation. From left to right, dynamical pictures for values of c ∈ (0.4, 1/4, −0.5).

1 The dynamical transition that occurs as c passes the threshold 4 is called a saddle-node bifur- 1 ? 1 cation. Saddle because in the case c = 4 , the point x = 2 can be seen as a “saddle point”, whose attracting or repelling properties depend on what side of the point you are looking. Node because 1 for c slightly less than 4 , the pair x+, x−, respectively repelling and attracting, is called in other contexts a “source” (out of which orbits leave) and a “sink” (into which orbits converge) and the pair is referred to as a “node”.

The period-doubling bifurcation through c = −3/4. So far, we have seen in the regime 1 −3 1 q 1 4 > c > 4 that Fc has two fixed points x± = 2 ± 4 − c, that x+ is repelling and x− is attracting. Math 207 - Spring ’17 - Fran¸coisMonard 6

1 At this point, we should also add an important observation that for 4 > c ≥ −2, the interval [−x+, x+] is stable under Fc, that is to say Fc([−x+, x+]) ⊂ [−x+, x+]. In particular, we see that if a seed is in this interval, its entire orbit must remain inside of it. 0 As c decreases through the value −3/4, |F (x−)| becomes greater than 1, then x− becomes a repelling fixed point. Since the [−x+, x+] is still stable, what type of dynamics is happening there ? The answer is that the point x−, when becoming repelling, gives birth to an attracting 2-cycle in the process, which will attract all orbits that lie inside (−x+, x+) (except the fixed orbit x−). 2 −3 We see this by studying Fc , whose number of real roots switches from 2 when c > 4 to 4 when −3 2 c < 4 . Indeed, Using the fact that fixed points of Fc are fixed points of Fc , we can factor Fc(x)−x 2 out of the polynomial Fc (x) − x and using the method of undetermined coefficients, establish that

2 2 2 2 Fc (x) − x = (x + c) + c − x = (Fc(x) − x)(x + x + c + 1)  1 3 = (x − x )(x − x ) (x + )2 + c + . + − 2 4

3 3 We see that if c + 4 , the last polynomial has no real roots and when c + 4 ≤ 0, then we obtain 0 1 q 3 two new fixed points, say x± = − 2 ± −c − 4 , these are the ones forming an attracting 2-cycle. This transition from an attracting fixed point to an attracting two-cycle is called a period-doubling bifurcation.

−3 −3 Figure 3: The period-doubling bifurcation. Left: c = −0.5 > 4 , right: c = −1 < 4 .

The period-doubling route to chaos. Before c reaches −2, this is certainly not the only bifurcation that occurs. In fact, period-doubling bifurcations will occur several times and lead to quite complicated-looking orbit. Analyzing this in detail would be a lost cause and would require computing roots of polynomials of arbitrarily high order. However, one graphical way of seeing 1 what happens is, for several values of c between 4 and −2, plot on the y-axis some of the iterates n {Fc (0)} for 100 ≤ n ≤ 150. We are dropping the first 100 terms, assuming that the “transitory” regime of the orbit will be gone by then, and that the orbit of 0 will have reached either convergence, or a periodic orbit. The plot obtained can be seen Figure4. Note that the choice of initial seed inside [−x+, x+] does not matter as in this regime, almost all orbits trapped in this interval have the same long-term behavior. Math 207 - Spring ’17 - Fran¸coisMonard 7

2 Figure 4: The bifurcation diagram of the orbits of x0 = 0 under the dynamics Fc(x) = x + c.

References

[A] A History of complex dynamics. From Schr¨oderto Fatou and Julia, by Daniel S. Alexander. Aspects of Mathematics, Vieweg 1994.1

[DK] Chaos and , the mathematics behind the computer graphics. Editors: Devaney and Keen.

[G] Complex Analysis, Theodore W. Gamelin. Undergraduate Texts in Mathematics, Springer.