Math 2030 Fall 2012 3 Definitions, Applications, Examples, Numerics 3.1 Basic Set-Up Discrete means “not continuous” or “individually disconnected,” dynamical refers to “change over time,” and system is the thing that we are talking about. Thus this course is about studying how elements in some ambient space move according to discrete time steps. The introductory material in this course will involve so-called one-dimensional systems. Nonetheless, we introduce the general notion of a Discrete Dynamical System since it will play a major role later. A Discrete Dynamical System consists of a set X (called the state space) and a map F ( ) (called the dynamic map) that takes elements in X to elements in X; that is, F ( ) : X· X. Given a point x X, the first iterate of x is defined by · → 0 ∈ 0 x1 = F (x0). The second iterate of x0 is nothing but the first iterate of x1, and is denoted 2 x2 = F (x1)= F F (x0) =: F (x0). Note that the power of F in the last expression is not a “power” in the usual sense, but the number of times F has been used in the iteration. The th process repeats, where the k iterate xk is given by xk = F (xk−1) = F F (xk−2) . = . = F F F(x ) ··· 0 k times = F k(x ) | {z0 } The orbit of the seed x is denoted by x is defined as the (ordered) collection of all of the 0 h 0i iterates of x0. Notationally, this means x = x ,x ,x ,... h 0i { 0 1 2 } = F k(x ): k = 0, 1, 2,... , { 0 } 0 where in the last set, F (x0)= x0 by convention. It is helpful to view an iterate x1 as the state that x0 moves to over one “discrete time th th step”. The k iterate xk is the state that x reaches at the k time step. The orbit x is 0 h 0i then all the states that x0 reaches at some time step. This is what is meant by a discrete dynamical system. By “ordered” we mean that the two sets a,b and b,a are considered as different sets { } { } if a = b. On the other hand, repeated elements in the set should be counted only once, 6 1 and so the sets a,a and a are considered as the same set. The following sets are also { } { } considered the same (with a = b): a,b,a , a,b,a,b,a,b,... and a,b . An orbit can be a finite set. 6 { } { } { } Exercise 3.1. (a) If an orbit x consists of only one element x = x , what is F (x )? h 0i h 0i { 0} 0 (b) Describe precisely when an orbit consists of only finite many elements. (c) Can the set 1, 2, 3, 1, 3 be an orbit for any dynamic map F : R R? Why or why not? { } → (d) Suppose an orbit of F ( ) is 1 = 1, 3, 5, 7, 9, 2, 4, 6, 8 . What is the orbit with seed 1 of the map F 2( )? What· abouth i F 3({)? } · · A very important and special kind of orbit is when it consists of only a single element. This can be characterized as a property of the dynamics. Definition 3.1. Given F : X X, a fixed point of F is a point x¯ X such that F (¯x) =x ¯. → ∈ The main focus of this course is the study of the asymptotic behavior of orbits, and a fixed point gives the simplest kind of behavior because it is always constant. The major underlying issue is to predict what happens to xk when k is very large, and how this depends on the seed x0 and perhaps other parameters. What kind of behavior might one expect? First, an observation: Observation 1. If y0 x0 , then the eventual behavior of y0 is the same as the eventual behavior of x . ∈ h i h i h 0i 3.2 Applications There is a myriad of applications of both continuous and discrete dynamical systems. We give two very simple ones here, and then give two more theoretical type applications on how discrete dynamical systems can arise in the development of algorithms. 3.2.1 Application in Finance One can describe the change in the amount of money in a bank account (= the state of the system) over time using the discrete dynamical systems. Suppose a bank offers annual interest at r% that is compounded monthly (with r> 0). If the initial amount is $ x0, then r r 2 the amount after 1 month is 1+ 12 x0, after two months 1+ 12 x0, and after k months k 1+ r x . In this case, we have F (x) = 1+ r x, and an explicit formula for the kth 12 0 12 k r k iterate is just F (x)= 1+ x. It is clear that xk as k . 12 →∞ →∞ 2 3.2.2 Application in Biology A crude way to predict the growth rate of a species is that it is proportional to the size of the population (the more species then the more that can reproduce). Described with discrete time, this means that the new population size after one time period is a multiple of the previous size, say xk+1 = λxk. Again we have a simple description of the entire system since k the size at the kth period is λ x0, where x0 is the initial size. In fact, this model is the same r as the financial model above if λ = 1+ 12 . 3.2.3 Newton’s method Discrete dynamical systems arise naturally in the construction of algorithms. For example, recall Newton’s Method of finding a root of an equation f(x) = 0 (this should have been covered in your Math 1550 class). One can approximate the rootx ¯ of the equation f(x) = 0 by beginning at some (hopefully) nearby point x0 and defining the iterates f(x0) xk+1 = xk 0 . − f (xk) Provided f 0(x) = 0 for all the x’s that come under consideration, this is an example of a discrete dynamical6 system. In this case, the iteration step is not naturally “time”as in the finance and biology examples above, but rather the number of times the algorithm has been iterated. As we study systems in general, it is often helpful to think of it as time nonetheless with each iterate taking one time step. Note that the function F ( ) in this case is given by · f(x) F (x)= x . (1) − f 0(x) Exercise 3.2. Consider the function f(x) = x2 2, which obviously has roots √2. Use x = 1 as a seed, approximate √2 by using F ( ) as− in (1) and by taking successive± iterates. 0 · How large must k be in order to have xk √2 < .0001? Work this out numerically, say, with Excel - see the next section). | − | 3.2.4 Euler’s method of discretizing differential equations Given a continuous function f( ) : R R and initial value x0 R, an ordinary differential equation has the form · → ∈ x˙(t)= f x(t) for t [0, ) ∈ ∞ (2) x(0) = x0, d wherex ˙(t) = dtx(t) is the time derivative of the solution x( ). The theory of Ordinary Differential Equations (ODEs) is an enormous and important subject· (you should be taking a course in it soon, say Math 2090). When f( ) is nonlinear (Math 2090 deals only with · 3 linear f( )) and continuous, solutions of (2) are known to exist, but can be very difficult if · not impossible to find concretely. A way to approximate a solution is to first fix a (small) parameter h> 0, and then successively “follow the dots” by using the discrete time step h: x1 = x0 + h f(x0) x2 = x1 + h f(x1) . = . xk+1 = xk + h f(xk) . = . We then interpolate these data points by defining h x (t)= xk +(t k h)xk when k h t< (k + 1)h (3) − +1 ≤ which is a piecewise linear function, called an Euler polygonal arc. The reason xh( ) should be “close” to an actual solution of (2) is because whenever kh<t< (k + 1)h, one· has h xk+1 xk x˙ (t)= f(xk)= − h which looks close to (2) if h is small (remember, the sequence xk also depends on h). Here is a picture of approximating the ODE where f(x) = x and x0 = .1. The solution in this case is easy to find, and is x(t)=(.1)et. 3.3 Examples In all the examples of this section, we consider a map F ( ) : R1 R1. Some of the · → Mathematica code that generates the graphics below is given in Section 3.5. 4 3.3.1 Linear Examples Actually the two applications of the previous subsection are special cases of so-called Linear Systems, in which F ( ) has the form · F (x)= ax + b (4) for some constant a and b. Exercise 3.3. Let F be as in (4). What are teh fixed points of F ? What is the general formula for F k(x)? (The answer should be given in terms of a and b). For what values of a k and b does F (x0) with x0 = 0 have a (finite) limit as k ? (You can peek ahead to the next subsection for all the answers, but it is worth trying→∞ to do yourself first.) It turns out that just about the only type of dynamics for which there exists an explicit formula for a general iterate are the linear or piecewise linear ones.