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1 Introduction We shall consider continuous maps f of an interval I into itself and the discrete dynamical system xn 1 f xn n 0 1 1 We apply 1 1 to an initial condition x0 to generate its orbit the sequence x0 x1 x2 A convenient way to visualize the orbit is to draw the cobweb diagram bouncing between the graph of f and the line y x see the figures below The orbit is generated by repeated composition of functions We denote k fold composition by f k that is f 2 x f f x f 3 x f f f x etc A fixed point or equilibrium is a solution to x 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 1 are distinct Clearly f p xk xk 0 k p 1 Graphically fixed points are given by the intersection between the graph of f and the line y x An intersection between the graph of f p and the line y x give points with period not greater than p We typically study questions of asymptotic behavior such as convergence to a fixed point or periodic orbit Let us note at the outset that the dynamics are trivial if f is monotone and I is bounded In this case the orbit is a bounded monotone sequence and every initial condition converges to a fixed point We thus consider unimodal maps ie maps with a single hump The fundamental example is the family of logistic maps f 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 the restriction to r 4 The orbit diagram of the logistic map has become an icon for chaos see Figure 1 1 This is a numerically generated picture that describes 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 bifurcations Linear maps f x ax admit the exact solution xn an x0 The origin is an attracting fixed point if a 1 and repelling if a 1 Observe that if a 0 the dynamics are oscillatory that is xk and xk 1 have opposite sign In particular there are two distinct forms of neutral stability if a 1 1 Figure 1 1 Orbit diagrams for the logistic and sine maps every point is fixed and if a 1 every orbit is of period 2 Linear maps are used to describe the stability of fixed points If x is a fixed point the characteristic multiplier at x is f 0 x If f 0 x 6 1 the fixed point is hyperbolic and the characteristic multiplies determines stability Exercise 1 We may expect bifurcations when f 0 x 1 In the logistic family the fixed point x r 1 1 r loses stability to a period 2 orbit in a period doubling bifurcation at r 3 This is illustrated graphically in Figures 1 2 1 4 When r 3 trajectories spiral into the fixed point The rate of approach is very slow as r approaches 2 as is seen from the density of orbits in Figure 1 3 When r 3 the fixed point has lost stability to period 2 orbit This corresponds to new fixed points of f 2 seen in the lower half of Figure 1 4 The period 2 orbit then loses stability in another period doubling bifurcation at r 1 6 Exercise 2 Trajectories when r is just below this bifurcation point are illustrated in Figure 1 5 These are the first two steps in a period doubling cascade at parameter values rn At this value an orbit with period 2n 1 loses stability to an orbit with period 2n in a period doubling bifurcation The first few terms in the 2 Figure 1 2 A stable fixed point with r 3 3 Figure 1 3 Slow approach to the fixed point as r 3 4 Figure 1 4 Birth of a stable period 2 orbit for r 3 5 Figure 1 5 Loss of stability of period 2 orbit near r r1 6 increasing sequence rn are 3 1 6 3 54409 3 5644 3 568759 The bifurcation values accumulate at r 3 566946 The orbit diagram of the logistic map include windows of chaos and chaotic bands beyond r 1 2 Universality and renormalization The most remarkable feature of the orbit diagram of the logistic is that its essential features depend only on minimal properties That is all families of the form rf1 x where f1 I I is unimodal with a nondegenerate maximum have similar orbit diagrams Roughly speaking this is what is meant by universality To illustrate this point Figure 1 1 compares the orbit diagram for the sine family f x r sin x r 0 1 and the logistic map In both orbit diagrams we see the same period doubling cascade and similar windows of order and chaos Metropolis Stein and Stein discovered that the ordering of these periodic orbits depends only on the fact that f is unimodal and continuous This is an example of qualitative or combinatorial universality A more dramatic quantitative feature is the following Let rn and r n denote the values of the period doubling bifurcations for the logistic and sine map respectively Numerical experiments reveal the amazing fact that lim r n 1 r n rn 1 rn lim 4 669201609102290 rn 1 n r n r n 1 n rn 1 4 The number is known as Feigenbaum s constant in honor of his penetrating analysis of this quantitative universality The central feature of Feigenbaum s analysis is the notion of renormalization To explain this idea we consider a family of maps f r x undergoing a cascade of period doubling bifurcation at the values rn We assume f r maps the interval I 1 1 into itself and has a critical point at 0 The characteristic multiplier of the orbit with period 2n decreases from 1 at rn to 1 at rn 1 We focus on the superstable orbit with characteristic multiplier 0 at the value Rn rn rn 1 It is easy to compute the values R0 and R1 for the logistic map Exercise 4 Let us compare the graphs of f R0 x Figure 1 6 and f 2 R1 x the lower half of Figure 1 7 The main idea is that by restricting the graph of f 2 R1 x to a smaller interval we again obtain a unimodal map More precisely the assumption that the period 2 orbit of f 2 R1 x is superstable 7 Figure 1 6 Superstable fixed point at r R0 8 implies that the interval 0 f R1 0 is positively invariant consider an orbit diagram in the upper half of Figure 1 7 Let 1 f R1 0 denote the length of this interval We restrict f 2 R1 to the interval 1 1 and rescale the x axis …