1 IntroductionWe shall consider continuous maps f of an interval I into itself and thediscrete dynamical systemxn+1= f(xn), n ≥ 0. (1.1)We apply (1.1) to an initial condition x0to generate its orbit, the sequence{x0, x1, x2, . . .}. A convenient way to visualize the orbit is to draw the cobwebdiagram, bouncing between the grap h of f and the line y = x (see th e figuresbelow ). The orbit is generated by repeated composition of functions. Wedenote k-fold composition by fk, that is f2(x) = f(f(x)),f3(x) = f(f(f (x)))etc. A fixed point or equilibrium is a solution tox = f (x). (1.2)A periodic orbit with minimal or prime period p is an orbit such that xk=xk mod p, and x0, x1, . . . , xp−1are distinct. Clearly, fp(xk) = xk, 0 ≤ k ≤p − 1. Graphically, fixed points are given by the intersection between thegraph of f and the line y = x. An intersection between the graph of fpandthe line y = x give points with period not greater than p.We typically study questions of asymptotic behavior, such as convergenceto a fixed point or periodic orbit. Let us note at the outset that the dynamicsare trivial if f is monotone and I is bounded. In this case, the orbit is abounded monotone sequence, and every initial condition converges to a fixedpoint. We thus consider unimodal maps (ie. maps with a single hu mp). Thefundamental example is the family of logistic mapsf(x) = rx(1 −x), r ∈ [0, 4]. (1.3)f takes a unique maximum value r/4 at the critical point x = 1/2, thus therestriction to r ≤ 4. The orbit d iagram of the logistic map has become anicon for chaos (see Figure 1.1). This is a numerically generated picture thatdescribes the asymptotic behavior of a typical initial condition as r is varied.The complexity of the orbit diagram for this simple example is striking.1.1 Stability and bifurcationsLinear maps f(x) = ax admit th e exact solution xn= anx0. The origin isan attracting fixed point if |a| < 1 and repelling if |a| > 1. Observe thatif a < 0 the dynamics are oscillatory, that is xkand xk+1have oppositesign. In particular, there are two distinct forms of neutral stability: if a = 11Figure 1.1: Orbit diagrams for the logistic and sine mapsevery point is fixed, and if a = −1 every orbit is of period 2. Linear mapsare used to describe the stability of fixed points. If x∗is a fi xed point, thecharacteristic multiplier at x∗is f0(x∗). If |f0(x∗)| 6= 1, the fixed point ishyperbolic and the characteristic multiplies determines stability (Exercise1). We may expect bifurcations when |f0(x∗)| = 1.In the logistic family, the fixed point x∗(r) = 1 −1/r loses stability to aperiod-2 orb it in a period-doubling bifurcation at r = 3. This is illustratedgraphically in Figures 1.2– 1.4. When r < 3 trajectories spiral into th e fixedpoint. The rate of approach is very slow as r approaches 2 (as is seen fromthe density of orbits in Figure 1.3). When r > 3 the fixed point has loststability to period-2 orbit. T his corresponds to new fixed points of f2seenin the lower-half of Figure 1.4.The period-2 orbit then loses stability in another period-doubling bifu r-cation at r = 1 +√6 (Exercise 2). Trajectories when r is just below thisbifurcation point are illustrated in Figure 1.5.These are the first two steps in a period-doubling cascade at parametervalues rn. At this value, an orbit with period 2n−1loses stability to an orbitwith period 2nin a period-doubling bifurcation. The first few terms in the2Figure 1.2: A stable fixed point with r < 3.3Figure 1.3: Slow approach to the fixed point as r → 3.4Figure 1.4: Birth of a stable period-2 orbit for r > 3.5Figure 1.5: Loss of stability of period-2 orbit near r = r1.6increasing sequence rnare3, 1 +√6, 3.54409 . . . , 3.5644 . . . , 3.568759, . . .The bifurcation values accumulate at r∞= 3.566946 . . .. The orbit diagramof the logistic map in clude windows of chaos and chaotic bands beyond r∞.1.2 Universality and renormalizationThe most remarkable feature of the orbit diagram of the logistic is that itsessential f eatures depend only on ‘minimal’ properties. That is all familiesof the form rf1(x) where f1: I → I is unimodal with a nondegeneratemaximum have similar orbit diagrams. Roughly speaking, this is what ismeant by universality.To illustrate this point, Figure 1.1 compares the orbit diagram for thesine family (f(x) = ˜r sin πx, ˜r ∈ [0, 1]) and the logistic map. In both orbitdiagrams we see the same perio d-doubling cascade, and similar windows oforder and chaos. Metropolis, Stein and Stein discovered that the orderingof these periodic orbits depends only on the fact that f is unimodal andcontinuous. This is an example of qualitative (or combinatorial) universality.A m ore dramatic quantitative feature is the following. Let rnand ˜rndenote the values of the period-doubling bifurcations for the logistic andsine map respectively. Nu merical experiments reveal the amazing fact thatlimn→∞rn+1− rnrn− rn−1= limn→∞˜rn+1− ˜rn˜rn− ˜rn−1= δ = 4.669201609102290 . . . (1.4)The number δ is known as Feigenbaum’s constant in honor of his penetratinganalysis of this quantitative universality.The central feature of Feigenbaum’s analysis is the notion of renormaliza-tion. To explain this idea, we consider a family of maps f (r, x) un dergoin ga cascade of period-doubling bifu rcation at the values rn. We assume f(r, ·)maps the interval I = [−1, 1] into itself, an d has a critical point at 0. Thecharacteristic multiplier of the orbit with period 2ndecreases from 1 at rnto−1 at rn+1. We focus on the superstable orbit with characteristic multiplier0 at the value Rn∈ (rn, rn+1). It is easy to compute the values R0and R1for the logistic map (Exercise 4).Let us compare the graphs of f(R0, x) (Figure 1.6) and f2(R1, x) (thelower half of Figure 1.7). The main idea is that by restricting the graphof f2(R1, x) to a smaller interval we again obtain a unimodal map. Moreprecisely, the assumption that the period-2 orbit of f2(R1, x) is superstable7Figure 1.6: Superstable fixed point at r = R0.8implies that the interval [0, f (R1, 0)] is positively invariant (consider an orbitdiagram in the upper half of Figure 1.7). Let α1= f(R1, 0) denote thelength of this interval. We restrict f2(R1, ·) to the interval [−α1, α1] andrescale the x-axis by the factor α1, and the y-axix by −α1to obtain a newunimodal map with a superstable fixed point at the origin. This operationis renormalization, and we denote it