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/