MATH34042: Discrete Time Dynamical Systems David Broomhead (based on original notes by James Montaldi) School of Mathematics, University of Manchester 2008/9
Webpage: http://www.maths.manchester.ac.uk/∼dsb/Teaching/DynSys email: [email protected]
Overview (2 lectures)
Dynamical systems continuous/discrete; autonomous/non-autonomous. Iteration, orbit. Applica- tions (population dynamics, Newton’s method). A dynamical system is a map f : S S, and the set S is called the state space. Given an initial point x0 S, the orbit of x0 is the sequence x0,x1,x2,x3,... obtained by repeatedly applying f: ∈ → n x1 = f(x0), x2 = f(x1), x3 = f(x2),...,xn = f (x0), ...
Basic question of dynamical systems: given x0, what is behaviour of orbit? Or, what is behaviour of most orbits?
+ + Example Let f : R R be given by f(x)= √x. Let x0 = 2. Then x1 = f(2)= √2 1.4142. And ≈ x = √2 1.1892 and x = √1.1892 1.0905. 2 p → 3 The orbit of≈2 therefore begins {2.0, 1.4142,≈ 1.1892, 1.0905, ...}. The orbit of 3 begins {3.0, 1.7321, 1.3161, 1.1472, ...}. The orbit of 0.5 begins {0.5, 0.7071, 0.8409, .9170, .9576, ...}. All are getting closer to 1.
Regular dynamics fixed points, periodic points and orbits. Example f(x)= cos(x). Globally attracting fixed point at x = 0.739085 . ···
Chaotic dynamics Basic idea is unpredictability. There is no “typical’ behaviour. Example f(x)= 4x(1 − x), x [0,1]: Split the interval into two halves: L = [0, 1 ] and R = [ 1 ,1]. We will ∈ 2 2 prove that given any sequence of Ls and Rs, say LLRRLRRLLL... there is an initial point x0 such that x0 L, x1 L, x2 R,... (see ‘itineraries’ in §2) ∈ ∈ ∈
Fractals We see relation between fractal sets and dynamics, and what it means to say the dimen- sion of a set is (say) 1.5.
Bifurcations How does dynamical system — or behaviour of orbits — depend on external param- eters. Eg fµ(x)= µx(1 − x) — how can dynamics change as µ changes?
0.1 §1 MATH34042: Discrete Time Dynamical Systems
1 Regular dynamics (mostly 1-dimensional)
As usual, f : S S is a dynamical system. Almost always, we assume S R. Given any x0 S, ⊂ ∈ then its orbit is {x0,x1,x2,...} where xn+1 = f(xn). →
1.1 Fixed points
A fixed point is one where f(p)= p.
Proposition 1.1 If xn p and f is continuous then p is a fixed point.
→ PROOF A function f is continuous at x if, whenever xn x then f(xn) f(x). So we have
lim f(xn)= f(p→) → xn p
→ but the sequence (xn)n=1 is exactly the same as (f(xn−1))n=1, so we have ∞ ∞ xn p f(xn−1) p and, therefore, f(p)= p as required. → ⇐⇒ →
Theorem 1.2 (Fixed point theorem) Let I be a compact1 interval and suppose f : I I is contin- uous. Then f has a fixed point. → The basic truth of this can be seen with a convincing picture. Imagine drawing a graph of f inside a square whose sides are I. The graphis a continuouscurve, consisting of points (x,y) where y = f(x). Since f(x) is defined for all values of x I, the graph must intersect the diagonal of the square at least once. However, on the diagonal x =∈y, so at such an intersection we have simultaneously y = x and y = f(x) and hence a fixed point. Counter-example on an unbounded interval: f(x)= x + 1 (f : R R) has no fixed points!
→ PROOF Define g(x)= f(x)− x. This function is continuous, since f is continuous. Now let us write I = [xmin,xmax]. We notethat since we havea dynamicalsystem f(xmin) I and so g(xmin) 0. ∈ ≥ Similarly, g(xmax) 0. It follows from the intermediate value theorem that there exists at least one point p I where g≤(p)= 0 and hence that there is at least one fixed point of f. ∈
1.2 Graphical analysis
Useful in 1-dimensional systems (S R). The basic idea is that suggested by the ‘convincing picture’ approach to the above fixed point⊂ theorem. Examples including f(x)= x2 on [0,1], f(x)= 3.0x(1 − x) on [0,1], and f(x)= x2 + x on R. Fixed points occur at intersection of graph with diagonal.
1for us, compact = closed and bounded
2008/9 1.2 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems
1 1 1
1 µ 4 b
0 0 0 p 0 1 0 1 0 1 f(x)= x2 f(x)=3.0x(1-x) f(x)= x2 + x
1.3 Stability of fixed points
Let f : S S have fixed point p S. We say p is attracting or asymptotically stable if ε > 0 such that ∈ ∃ → dist(x0,p) < ε = xn p and dist(xn,p) < ε n > 0 ∀ R Rk Here ‘dist(x0,p)’ is the distance between⇒ x→0 and p. If S then dist(a,b)= |a − b|; if S then dist(a,b)= a − b = (a − b )2 + + (a − b⊂k)2. (For metric space fans, ‘dist’ is⊂ any p 1 1 k metric—we will ink practicek stick to R in this··· course.)
A point p is repelling if ε > 0 s.t. dist(x0,p) < ε (x0 = p) implies there is an n > 0 for which ∃ 6 dist(xn,p) > ε. (If you are interested in reading more about the various concepts of stability Milnor has a nice, short article in Scholarpedia, see: http://www.scholarpedia.org/article/Attractor)
Example if f : R R is linear, so f(x)= ax for some a, then 0 is a fixed point. It is attracting if |a| <1 and repelling if |a| >1. → n We can solve the linear difference equation explicitly: xn = a x0. Letting |a| <1, and choosing any n ε>0, then |x0| <ε = |xn| = |a ||x0| <ε and limn xn = 0. So x = 0 is an attracting fixed point in this case. A similar argument shows that x = 0 is repelling if |a| >1. ⇒ →∞ Given ε > 0, write Nε(p)= {x S ||x − p| < ε}. Called the ε-neighbourhood of p. ∈ Beware: If S = [0,1], then Nε(0) = [0,ε) (not (−ε,ε), as x < 0 is not in S). If f : I I (I R) is differentiable, then a fixed point is hyperbolic if |f (p)| = 1. ⊂ ′ 6 2 Example→ in f(x)= x + x, the origin is a fixed point which is not hyperbolic, as f′(0)= 1.
Theorem 1.3 (Stability theorem) Suppose f : I I is differentiable and f′ is continuous (ie f is of class C1) and suppose p is a hyperbolic fixed point. Then → (a) |f′(p)| < 1 implies p is attracting, (b) |f′(p)| > 1 implies p is repelling.
If f (p)= 1 one needs a finer analysis (using eg second or higher order Taylor series). ′ ±
PROOF (Outline) (a) Let |f (p)|
That is, |f(x)−f(p)| 2008/9 1.3 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems And similarly |f3(x)− p| Example f : R R, f(x)= cosx has a unique fixed point p (in fact p 0.739085). Now |f′(x)| = |sinx| 1, and since 0 Basins of attraction If p is an attracting fixed points, it is natural to ask which initial conditions provide orbits that converge to p. The basin of attraction of p is the set of all these initial conditions: B(p)= {x0 S | xn p}. ∈ These basins of attraction can be very complicated in genera→l, but there is one general property: they are always open. Proposition 1.4 The basin of attraction of any attracting fixed point of a continuous dynamical system is open. PROOF Uses property that if f : S S is continuous and U S is open, then f−1(U) is open. ⊂ → 1.4 Stability of periodic orbits Dynamical system: f : S S. A periodic point of period k > 0 is a point p S for which the condition fk(p)= p is true2. That is, p is a fixed point of fk. Note: if p has period∈k, then it also has period 2k, 3k etc. The prime→ period (also, primitive or fundamental period) is the smallest positive period k. That is, it’s the smallest k such that xk = x0. If p is a periodic point of period k, then the sequence p,f(p),...,fk−1(p) is a periodic orbit. Example f(x)=−x3 has three periodic points: a (attracting) period 1 point (ie, fixed point) at x = 0 and a (repelling) period 2 point at x = 1; the corresponding periodic orbit is {1,−1}. And the tent map 8 2 4 T :[0,1] [0,1] (see introduction) has a period 3 point at x = 8/9, with periodic orbit { 9 , 9 , 9 }. Useful fact→ — follows from chain rule — about the derivative of fk: k (f )′(x0)= f′(x0) f′(x1) f′(x2) f′(xk−2) f′(xk−1). (1) · · ··· · Corollary 1.5 (of Theorem 1.3) If f : I I is class C1, and p is a period k point, then the periodic k k orbit is attracting if |(f )′(p)| < 1 and repelling if |(f )′(p)| > 1. → Stability of periodic orbit: p0,...,pk−1 is p.o. There is ε > 0 s.t. if |x0 − p0| < ε then sequence: x0,xk,x2k,...xnk,... p0 − it then follows that x1,xk+1,x2k+1,...xnk+1,... p1 etc.→ Example For f(x)= x3, f (0)= 0 so it’s attracting,→ and f ( 1)= 3, so |f (1)||f (−1)| = 9 and the ′ ′ ± ∓ ′ ′ periodic orbit is repelling. For the tent map, let |T ′(x)| = 2 (provided x = 1/2), so fixed points and periodic orbits are all repelling. ± 6 2fk is a notation indicating iterations of the map in the sense that f2(x) = f(f(x)) and generally fk(x) = f(fk−1(x)) 2008/9 1.4 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems An attracting periodic orbit also has a basin of attraction: B = x0 S | xn {p0,p1,...,pk−1} . ∈ → Again B is an open subset of S — with the same proof as for Proposition 1.4. Remark If p is an attracting fixed point with |f′(p)| = a the the smaller a is the faster the convergence to 2 2 p. This is because, if x = p + h then f(x0) p + ah (Taylor’s theorem) and f (x0)= p + a h etc, so n n ≈ |f (x0)− p| a |x0 − p|. ≈ Now do a better analysis valid when a = 0, using f(p + h) p + bh2. We can use this to derive an dynamical system which shows how the small perturbation≈ from the fixed point, h, evolves. 2 Assume that xn = p + hn, then xn+1 = p + hn+1 = f(p + hn) p + bh . So we have to solve ≈ n 2 hn+1 = bhn given the initial condition h0 = h. We can solve this equation explicitly: 1 n h = (bh)2 n b (Easy proof by induction that this is indeed the solution.) It is clear that if |bh| < 1, the sequence (hn) converges to zero extremely rapidly, that is, faster than exponentially. To see this note that if |bh| < 1, the sequence (|bh|n) converges to zero exponentially, but here, instead of the exponent being n, it is 2n. In the dynamical systems context this kind of stability is sometimes referred to as superstability. Example The convergence properties of Newton’s method are an example of this phenomenon. Let us say we wish to find the zeros of a polynomial g(x). Newton’s method is to iterate the following difference equation: def g(xn) xn+1 = f(xn) = xn − g′(xn) Clearly any p which satisfies g(p)= 0 is a fixed point of this dynamical system. If we now calculate f′(p) we find g′(p) 1 f′(p)= 1 −( + g(p)( )′)= 0 g′(p) g It follows that if you begin sufficiently close to a root of g, Newton’s method give faster than exponential convergence to the actual root. 1.5 The Logistic family and the period doubling cascade. This family of maps (or dynamical systems) is important for two reasons: firstly it shows how complex behaviour can result from simple ingredients, and secondly because of it its historical role. fµ : [0,1] [0,1] − x µx(1 − x) 7−→ → For this to be a dynamical system requires 0 µ 4. ≤ ≤ 2008/9 1.5 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems 1 1 1 1 µ 4 b 0 0 p 0 0 1 0µ 1 0 1 0<µ<1 1<µ<4 µ = 4 Fixed points Solve fµ(x)= x : 1 µx(1 − x)= x x = 0 or 1 − . µ 1 ⇐⇒ 1 So fixed points are x = 0 and x = 1 − . Put pµ = 1 − . [Note: pµ [0, 1] iff µ > 1.] µ µ ∈ Question: when are these attracting/repelling? Use the Stability theorem (Theorem 1.3): fµ′ (0)= µ, so 0 is attracting if µ [0,1) and repelling if µ (1,4]. (What happens for µ = 1? Need finer analysis — graphics?) ∈ ∈ Transition occurs at µ = 1—call this value µ1. fµ′ (pµ)= 2−µ, so pµ is attracting if µ (1,3) and repelling if µ (3,4]. Again, need finer analysis for µ = 3. ∈ ∈ This next transition occurs for µ = 3 — call this µ2. So far: For 0<µ<µ1 0 is a global attractor: B(0) = [0,1] (easy to see as all sequences are monotone decreasing). For µ1 <µ<µ2 every x0 has an orbit converging to pµ, except of course x0 = 0 (which is a fixed point) and x0 = 1 (which maps to 0, so is ‘eventually fixed’). So, B(pµ) = (0,1) for µ (1,3). ∈ 0 is attracting pµ is attracting ?? µ 0 µ1 = 1 2 µ2 = 3 4 For µ > µ2 both fixed points are repelling, so what happens? Try on calculator: 2008/9 1.6 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems x0 = 0.4 x1 = 0.744 x2 = 0.590438 x3 = 0.749646 x4 = 0.581798 x5 = 0.754257 x6 = 0.574596 x7 = 0.757751 x8 = 0.569050 x9 = 0.760222 x10 = 0.565082 x11 = 0.761868 x12 = 0.562418 x13 = 0.762924 x = 0.560699 x = 0.763580 µ = 3.1 14 15 x16 = 0.559630 x17 = 0.763976 x18 = 0.558983 x19 = 0.764216 x20 = 0.558589 x21 = 0.764361 x22 = 0.558351 x23 = 0.764446 x24 = 0.558211 x25 = 0.764494 x26 = 0.558133 x27 = 0.764523 x28 = 0.558085 x29 = 0.764539 x30 = 0.558059 x31 = 0.764549 Seems to be converging(albeit slowly) to a period-2 orbit, {q0,q1} with q0 0.558 and q1 0.764. 2 ≈ ≈ (And in fact period-2 orbit has (f )′(q)= 0.59). Try another value of µ: x0 = 0.4 x1 = 0.792 x2 = 0.543628 x3 = 0.818718 x4 = 0.489781 x5 = 0.824655 x6 = 0.477175 x7 = 0.823280 x8 = 0.480115 x9 = 0.823695 x = 0.479230 x = 0.823576 µ = 3.3 10 11 x12 = 0.479484 x13 = 0.823611 x14 = 0.479409 x15 = 0.823600 x16 = 0.479432 x17 = 0.823604 x18 = 0.479425 x19 = 0.823603 x20 = 0.479427 x21 = 0.823603 x22 = 0.479427 x23 = 0.823603 2 Convergence is now faster (and attracting period-2 orbit has (f )′(q)=−0.30): Period 2 points These are found by solving f(f(x)) = x: 2 2 fµ(fµ(x)) = µ(fµ(x))(1 − fµ(x)) = = µ x(1 − x)(1 − µx + µx ). ··· 2 1 So fµ(x)= x is a degree 4 equation. We know already that x = 0 and x = pµ = 1 − µ satisfy this equation (as they are fixed points), so take out those factors and there remains a quadratic factor, with just 2 solutions: 2 2 2 fµ(x)− x = −x(µx − µ + 1)(µ x + bx + (µ + 1)). [Now you find b.] The roots are 1 q = µ + 1 (µ + 1)(µ − 3) . ± 2µ ± p So these are two period-2 points, together forming a single period-2 orbit: you can check that fµ(q+)= q− and vice versa. 2 Important: Notice that as µ µ2 from above, so q , which is precisely the fixed point pµ for ± 3 µ = µ2 = 3. → → 2008/9 1.7 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems What about their stability? Use Corollary 1.5. Write ∆ = (µ + 1)(µ − 3): 2 (f )′(q+) = f′(q+)f′(q−) = µ(1 − 2q+)µ(1 − 2q−) = (µ − 2µq+)(µ − 2µq−) = (1 − √∆)(1 + √∆)= 1 − ∆ = 4 + 2µ − µ2. So, the period 2 orbit is hyperbolic provided 4 + 2µ − µ2 = 1. 6 ± 4 + 2µ − µ2 = 1 µ = 3 4 + 2µ − µ2 =−1 µ = 1 + √6 3 45 ⇒ ≈ · This new transition⇒ occurs at µ = µ3 := 1 + √6: for µ2 <µ<µ3 the periodic orbit is attracting, while for µ > µ3 it is repelling. For µ2 <µ<µ3 the basin of attraction B({q+,q−})is a little complicated to describe. Of course 0,1,pµ are not in it, but nor are all points that eventually map to pµ, and in particular the symmetric point to pµ (which is 1 − pµ). Also the two points mapped to 1 − pµ, and the two points mapped to each of those and so on. So the basin is a union of lots of (infinitely many in fact) open intervals. Picture so far: q is attracting 0 is attracting pµ is attracting ± ?? µ 0 µ1 = 1 2 µ2 = 3 µ3 4 Let’s look in detail at what happens near pµ as µ passes through the value 3. 2 y = fµ(x) q+ y = fµ(x) pµ pµ pµ 2 y = fµ(x) q− 0.5µ = 2.9 0.8 0.5µ = 3 0.8 0.5µ = 3.1 0.8 2 Notice how the graph of fµ intersects the diagonal once for µ = 2.9(< µ2), becomes tangent at µ = 3(= µ2)and then intersects it three times for µ = 3.1(> µ2) 2008/9 1.8 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems Recap: For µ1 < µ µ2, the fixed point pµ is attracting. ≤ As µ passes µ2 =3, pµ loses stability (becomes repelling) and there is an attracting period-2 orbit born. It turns out this is typical behaviour as the derivative f′(pµ) passes through -1, and is called a period doubling bifurcation. Bifurcation diagram: x 1 q+ pµ q− 0 µ 2 µ1 = 3 µ2 4 This diagram shows how the period-2 orbit bifurcates from the fixed point (when µ = 3). The continuous curves represent where the fixed or periodic points are attracting, and the broken curves where they are repelling. After period 2 is unstable (µ > 1 + √6 3.45). Period 2 orbit still exists but is repelling. So what happens to an arbitrary initial point? ≈ µ = 3.5 x0 = 0.4 x1 = 0.840 x2 = 0.4704000 x3 = 0.8719334 x4 = 0.3908294 x5 = 0.8332863 x6 = 0.4862208 x7 = 0.8743356 x8 = 0.3845551 x9 = 0.8283538 x10 = 0.4976432 x11 = 0.8749805 x12 = 0.3828637 x13 = 0.8269768 x14 = 0.5008016 x15 = 0.8749980 x16 = 0.3828177 x17 = 0.8269391 x18 = 0.5008879 x19 = 0.8749974 x20 = 0.3828193 x21 = 0.8269407 x22 = 0.5008841 x23 = 0.8749971 x24 = 0.3828201 x25 = 0.8269408 x26 = 0.5008840 x27 = 0.8749973 x28 = 0.3828196 x29 = 0.8269409 x30 = 0.5008837 x31 = 0.8749973 x32 = 0.3828196 x33 = 0.8269409 x34 = 0.5008837 x35 = 0.8749973 It seems to converge to a period 4 orbit. And indeed it does! And to an attracting period-4 orbit (necessarily attracting or you wouldn’t see convergence!). See figure below. 2008/9 1.9 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems As µ descends to µ2 = 1 + √6 this period-4 orbit converges to the period-2 orbit. In the figure, you would see the two ‘square loops’ get closer and then become a single ‘square loop’. So as µ passes through µ2 the period-2 orbit loses stability and an attracting period-4 orbit appears. 2 If we consider fµ, then each of q+ and q− are fixed points which for µ < µ2 are attracting, they then 2 lose stability as µ passes through this bifurcation value (determined by putting (f )′(q+)=−1) and 2 a period-2 orbit of fµ appears near each of q+ and q− : period doubling again! So for µ > µ2 we have a period 4 orbit, which starts off as attracting. But as µ passes through the value µ3 = 3.54409... it loses stability, and a period-8 orbit appears. Another period doubling bifurcation! And so it continues: µ1 = 3 period 2 orbit is born from period 1 point µ2 = 3.449... period 4 orbit is born from period 2 orbit µ3 = 3.54409... 8 µ4 = 3.5644... 16 µ5 = 3.568759... 32 . . . . µ = 3.569946... Called∞ a period doubling cascade∞ . It is not unusual. Note that the bifurcations occur closer and closer together — the separations are approximately in geometric progression. With common ratio δ 4.669... ≈ (often called the Feigenbaum number). What happens for µ > µ ? Well, one certainly has an period-2 orbit, a period-4 orbit, a period-8 orbit and so on, all of which are repelling! What about other periods? ∞ 2008/9 1.10 February 11, 2009 §1 MATH34042: Discrete Time Dynamical Systems Period 3 orbits These appear ‘out of the blue’ when µ = 3.828427... (ie, not bifurcating from something else). In fact there are 2 of them, and at first (µ just greater than 3.828427...) one is attracting and the other repelling. Later (larger µ) they are both repelling. 3 Graphs of fµ(x) for µ = 3.8 and µ = 3.85: µ = 3.8 µ = 3.85 3 At the bifurcation value µ = 3.828427... the graph of fµ is tangent to the diagonal at the three points. This bifurcation where two fixed points appear (here of f3 but it could be of f or f2 or whatever) is called the tangent bifurcation—also the saddle-node bifurcation. We will study this in detail in the final chapter, on bifurcations. We will prove (next chapter) that when µ = 4 there are periodic orbits of every period, and they are all repelling! 2008/9 1.11 February 11, 2009