Summary of Unit 1: Iterated Functions 1 Summary of Unit 1: Iterated Functions David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 2 Functions • A function is a rule that takes a number as input and outputs another number. • A function is an action. • Functions are deterministic. The output is determined only by the input. x f(x) f David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 3 Iteration and Dynamical Systems • We iterate a function by turning it into a feedback loop. • The output of one step is used as the input for the next. • An iterated function is a dynamical system, a system that evolves in time according to a well-defined, unchanging rule. x f(x) f David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 4 Itineraries and Seeds • We iterate a function by applying it again and again to a number. • The number we start with is called the seed or initial condition and is usually denoted x0. • The resulting sequence of numbers is called the itinerary or orbit. • It is also sometimes called a time series or a trajectory. • The iterates are denoted xt. Ex: x5 is the fifth iterate. David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 5 Time Series Plots • A useful way to visualize an itinerary is with a time series plot. 0.8 0.7 0.6 0.5 t 0.4 x 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 time t • The time series plotted above is: 0.123, 0.189, 0.268, 0.343, 0.395, 0.418, 0.428, 0.426, 0.428, 0.429, 0.429. David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 6 Fixed Points • A fixed point of a function is a number that is not changed when the function acts on it. • x is a fixed point if f(x)= x. 2 • Example: 1 is a fixed point of f(x)= x , because f(1) = 1. David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 7 Stability • A fixed point is stable if nearby points move closer to the fixed point when they are iterated. • A stable fixed point is also called an attracting fixed point or an attractor. • A fixed point is unstable if nearby points move further away from the fixed point when they are iterated. • An unstable fixed point is also called a repelling fixed point or a repellor. David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 8 Stability Stable Attracting Unstable Repelling • We are usually interested in stable behavior, as this is what we’re most likely to observe. David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 9 Phase Lines • The phase line lets us see all at once the long-term behavior of all initial conditions. • Time information is not included in the phase line. We can tell what direction orbits go, but not how fast. • Example: phase line for the squaring function: 0 1 • Seeds larger than 1 tend toward infinity • 1 is an unstable fixed point • Seeds between 0 and 1 approach 0 • 0 is a stable fixed point David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 10 Dynamical Systems • One of the goals of the study of dynamical systems is to classify and characterize the sorts of behaviors seen in classes of dynamical systems. • So far, for iterated functions, we have seen: – Fixed points (stable and unstable). – Orbits can approach a fixed point. – Orbits can tend toward infinity—keep growing larger and larger. – Orbits can tend toward negative infinity—keep growing more and more negative. • We will encounter more types of behavior soon... David P. Feldman http://www.complexityexplorer.org/.