Discrete Dynamical Systems Trinity College { MATH 210-01 October 19, 2015 1 Discrete Dynamical Systems MATH 210-01 October 19, 2015 One-Dimensional Discrete Dynamical Systems Suppose we invest P0 = $1000 into a savings account that accrues 1% interest annually. How much many do we have after the nth year? To find this solution, we first consider how much money we have after the first year. It is easy to see that we'll have P1 dollars given by P1 = P0 + 0:01P0 = (1 + 0:01)P0 = 1:01P0 = (1:01)(1000) = $1010:00: Once we have the amount for the first year, we can find the amount that we will have after two years. We simply add 1% of P1 to the amount P1. This gives P2 = P1 + 0:01P1 = (1 + 0:01)P1 = (1:01) (1:01)P0 2 = (1:01) P0 = (1:01)2(1000) = $1020:10: If we consider the boxed terms above, we can see that after the nth year, we may expect to have n Pn = (1:01) P0: It is easy to see that as n ! 1, the amount of money we have in the account, call it Pn, will approach infinity. In fact, this amount will grow exponentially. This process described above is an example of a discrete dynamical system because the amount of money (our quantity of interest) is changing by a determined amount based on the previous time step which is updated every year (a discrete update). We note that a dynamical system is nothing more than a sequence of numbers. In fact, it is a sequence of numbers (which may represent population totals, monetary amounts, concentrations, etc) that evolve in discrete time steps. In the previous example, the sequence of numbers we are interested in is the amount after every year: f$1000:00; $1010:00; $1020:10;:::g : Written another way, we see that our sequence of interest is simply an iterative process that involves multiplying by 1:01 after every year: 2 n P0; (1:01)P0; (1:01) P0;:::; (1:01) P0;::: : We can generalize any dynamical system in the following way: Take a point x in the real number system (which was P0 in the previous example) and apply a function to it, say f(x) (which was 2 Discrete Dynamical Systems MATH 210-01 October 19, 2015 (1:01)P0 in the previous example), then the dynamical system is the sequence of numbers defined as a repeated composition of that function: x; f(x); ff(x); f ff(x) ;::: Notation: x; f(x); f 2(x); f 3(x); : : : ; f n(x);::: : Each element of this sequence is in R, and we are interested in the limiting value of this sequence. If x0 is the initial point, we want to know if the limit n lim f (x0) n!1 exists or not. This limit may depend on the initial point x0. Essentially, a dynamical system is described by a function and its infinite iterations of composition of that function. The behavior of the system can be described by determining the above limit for all possible values of x0. Consider the interest problem again. In this example, our function is f(x) = 1:01x and our th initial point was x0 = 1000. We can find the n iteration as 0 n = 0 : f (x0) = x0 1 n = 1 : f (x0) = f(x0) = 1:01x0 2 2 n = 2 : f (x0) = f f(x0) = 1:01(1:01x0) = (1:01) x0 . n n n = n : f (x0) = (1:01) x0; which is precisely what we found before. Now, we can describe the behavior by looking at the following limit: n n lim f (x0) = lim (1:01) x0 = ±∞; n!1 n!1 which depends on the sign of x0. If x0 < 0 (which is impractical for the interest rate problem), the limit is −∞; and if x0 > 0, the limit is 1. In the case when x0 = 1000, we obtain positive infinity. In most cases, we will want to determine this limiting value for different starting positions x0. Another way to write a dynamical system is to display it as a recursive relationship. For example, some sequences are written recursively like the Fibonacci Sequence, f1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89;:::g; which can be written as x0 = 1; x1 = 1; xn+1 = xn + xn−1: Here, the \next" term is defined as an expression in the previous two terms. For the interest problem, we can simply write xn+1 = 1:01xn; x0 = 1000; which describes the same system as before. We note that the function f(x) = 1:01x is preserved in this notation as simply the right hand side of the equation. We will be working with the above notation for dynamical systems. We've already come across a dynamical system of this form called Newton's Method for finding roots of functions. 3 Discrete Dynamical Systems MATH 210-01 October 19, 2015 Newton's Method is an iterative, recursive process designed to find the root of a specified function, say g(x). We've seen this sequence defined as g(xn) xn+1 = xn − 0 ; g (xn) which starts at some initial point x0 and converges (hopefully) to a root of the function g(x), i.e., as n ! 1, xn ! r, where r is the solution to the equation g(r) = 0. We see here that the dynamical system that is Newton's Method is the system described by the g(x) function f(x) = x − g0(x) , where g(x) is the function whose roots we desire. In this sense, given an initial point x0, we can write the sequence 2 n fx0; f(x0); f (x0); : : : ; f (x0);:::g; which we know converges (under appropriate conditions) to the root, say x = r. That is, n lim f (x0) = r; n!1 g(x) where f(x) = x − g0(x) : Here, we may need x0 to be sufficiently close to r to begin with as we've seen various results occur for different values of x0. The point is, Newton's Method is a specific dynamical system whose behavior can be characterized as converging to a root of a specified function assuming our initial point is \close enough" to the desired root. This is a very important concept that can be generalized to any dynamical system. The key difference between Newton's Method and the interest rate problem is that for most values of x0 using Newton's Method, the sequence actually converges to some finite value. In the interest problem, the only starting value that doesn't approach ±∞ as we repeatedly apply f(x) = 1:01x is x0 = 0. If x0 = 0, then the system just stays at 0 forever, i.e. n n lim f (x0) = lim (1:01) (0) = 0: n!1 n!1 In the next section, we classify the dynamics of systems by considering what happens for any starting value x0. Trajectories and Classifications of Points So far, we have been investigating certain dynamical systems to determine what happens to an initial point x0 when we repeatedly apply f to it. That is, if we are given a function f(x) and an initial point x0, then we call the sequence 2 n fx0; f(x0); f (x0); : : : ; f (x0);:::g; a single trajectory of the dynamical system and we want to determine what happens to this trajec- tory by considering the following limit: n lim f (x0): n!1 In some cases (as in the Newton's Method case), this limit converges to a finite number. It turns out that this limit converges to what is called a fixed point. A fixed point is a specific type of trajectory. 4 Discrete Dynamical Systems MATH 210-01 October 19, 2015 Definition If f is a function and f(c) = c, then c is a fixed point. By definition, a fixed point is an initial value x0 that satisfies f(x0) = x0. In this case, the trajectory (or sequence) 2 n fx0; f(x0); f (x0); : : : ; f (x0);:::g; is the same as the sequence fx0; x0; x0; : : : ; x0;:::g: 2 We can see this is true because f(x0) = x0, f (x0) = f f(x0) = f(x0) = x0, and so on. A fixed point never moves, hence it's name. If we look at the function f(x) = x3, it follows that 0, −1, and 1 are fixed points. We determine this by setting f(x) = x and solving for x. In this case, we have f(x) = x ) x3 = x ) x3 − x = 0 ) x(x2 − 1) = 0 ) x(x − 1)(x + 1) = 0 ) x = −1; 0; 1: Graphically, we can say that a fixed point is the point at which the graph of y = f(x) intersects the line y = x. Consider the following figure: Fixed Points of f(x) = x3 1.5 1 0.5 0 y = f(x) −0.5 −1 −1.5 −1 −0.5 0 0.5 1 x From the figure it is easy to see that the fixed points of the system occur when x0 = −1; 0; 1. Another type of trajectory is called a periodic point or a cycle. Suppose we start with x0 and th we apply f to it many times. If the k iteration comes back to x0, then x0 is a periodic point with period k. A cycle is defined formally below. Definition The point c is periodic point or cycle if f k(c) = c for some k > 0 The smallest such k is called the prime period of the trajectory.