Dynamics, Chaos, and Fractals (part 1): Introduction to Dynamics (by Evan Dummit, 2015, v. 1.07) Contents 1 Introduction to Dynamics 1 1.1 Examples of Dynamical Systems . 1 1.2 Orbits, Fixed Points, and Cycles of Periodic Points . 3 1.2.1 Denitions and Examples . 3 1.2.2 General Existence of Fixed and Periodic Points . 6 1.2.3 The Doubling Function, the Logistic Maps, and Computational Diculties . 7 1.3 Qualitative and Quantitative Behavior of Orbits . 9 1.3.1 Orbit Analysis Using Graphs . 9 1.3.2 Attracting and Repelling Fixed Points . 11 1.3.3 Attracting and Repelling Cycles . 13 1.3.4 Weakly Attracting and Weakly Repelling Points (and Cycles) . 15 1.3.5 Basins of Attraction . 19 1.4 Newton's Method . 23 1 Introduction to Dynamics In this chapter, our goal is to provide an introduction to (discrete) dynamical systems on the real line. We begin by introducing some examples of more general dynamical systems, and then restrict our attention to discrete dynamical systems on R, which arise by repeatedly iterating a function f dened on a subset of the line. We study the structure of the orbit of a point under a function, and how to classify the behavior of xed points and cycles in terms of whether they attract or repel nearby points as we iteratively apply f. We will then discuss Newton's method, a procedure often familiar from calculus that provides a way to compute zeroes of dierentiable functions numerically, both because it is a computational aid and because it provides another source of interesting dynamical systems. 1.1 Examples of Dynamical Systems • Vaguely speaking, a dynamical system consists of two things: a space, and an operator that acts on that space. • In a discrete dynamical system, the operator can be thought of as a transformation of the space. The goal is to describe the behavior of points in the space as we iterate the transformation repeatedly. ◦ Example: The space is R and the transformation is the function f(x) = cos(x). What happens as we apply f repeatedly to a particular value of x (e.g., x = 2π)? ◦ Example: The space is R and the transformation is the function f(x) = x2 − 1. What happens as we apply f repeatedly? z + i ◦ Example: The space is and the transformation is the function f(z) = . What happens as we C z − i apply f repeatedly? ◦ Example: The space is R3 and the transformation is the map T (x; y; z) = (x + 2y; sin(z); cos(x)). What happens as we apply T repeatedly? 1 • In a continuous dynamical system, the operator can be thought of as a set of rules that tell the system how to change (or evolve) over time. ◦ Any dierential equation, such as f 00(t) = e2tf(t) + f 0(t), gives a dynamical system (in the sense above) the goal is to solve the equation, or at least describe its solutions. ◦ Another example: consider a solar system containing 2 planets with some given initial positions and velocities. As time goes forward, what kind of motion will the planets have? (This is called the 2-body problem, and it was essentially solved by Newton.) ◦ More generally, consider a solar system with n planets and given initial information. What kind of motion will the planets have? (This is called the n-body problem, and, perhaps surprisingly, its solutions can behave very unpredictably, unlike the solutions to the 2-body problem.) ◦ Earth's climate is another example of a dynamical system: it is well known that predicting the weather into the future with high accuracy is very computationally dicult. ◦ Similar computational diculties often arise in many other common models of physical phenomena. Since any physical phenomenon that evolves over time gives rise to a dynamical system, it is worthwhile to study abstract properties of dynamical systems. • We will see later that even simple-seeming dynamical systems can exhibit extremely complicated and unpre- dictable chaotic behavior. But we will give a taste of what is to come by analyzing the dynamics of a few simple population models: • Example (population model 1): A population of cats lives on an extremely large desert island with plentiful food. When the population is small, the cats can essentially breed with no restrictions. If the population is currently P , then the population one year later will be 4P (with one pair of cats producing eight ospring per year on average). ◦ In this case, the population at arbitrary number of years later can be found by iterating the function f(P ) = 4P : f(f(P )) = 16P , f(f(f(P ))) = 64P , and so forth. ◦ Thus, if the population at year 0 is 2 cats, then after n years, there will be 4n · 2 cats: i.e., we observe exponential population growth. ◦ If we change the starting population or the growth parameter slightly, the system will behave very predictably over time: we will always get exponential growth (at least, assuming the growth constant actually makes the population increase). • Example (population model 2): On another much smaller desert island there is also a population of cats. Since this island is smaller, when the population grows suciently large the cats will begin to compete for resources and breed more slowly (or even decrease in population, if there are more cats than the island can sustain). After careful study, it is determined that if the population is currently P , then the population one P year later will be 3:74P 1 − . 1 000 000 ◦ In this case, the population at arbitrary number of years later can be found by iterating the function P f(P ) = 3:74P 1 − . 1 000 000 ◦ Here are the results of a computer simulation for an initial population of 2 cats, and of 4 cats: 1´106 1´106 800 000 800 000 600 000 600 000 400 000 400 000 200 000 200 000 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 2 ◦ After a few generations of nearly-exponential growth, the population for a starting population of 2 cats seems to bounce around randomly for about 170 generations, but then settles into an extremely stable pattern that oscillates between ve population values which are (in order of how they appear from year to year) equal to 227476, 657233, 842539, 496175, and 934945. ◦ For 4 cats, the behavior is very similar (and the cycling population values are the same), but the pattern stabilizes much more quickly, after about 30 generations. • Example (population model 3): On a very slightly dierent desert island to the one considered above, the cats breed a very tiny bit more rapidly: if the current population is P , then the next year's population is P 3:75P 1 − . 1 000 000 ◦ Here are the results of a computer simulation for an initial population of 2 cats, and of 4 cats: 1´106 1´106 800 000 800 000 600 000 600 000 400 000 400 000 200 000 200 000 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 ◦ Unlike the previous example, the populations both appear to behave in a much less predictable manner. There are some runs where the populations almost cycle between a small number of values, but they are not stable and degenerate into seemingly random behavior quite rapidly. ◦ Later, we will study the dynamics of these types of maps, and give some explanations for the radically dierent behaviors of these two seemingly similar models. 1.2 Orbits, Fixed Points, and Cycles of Periodic Points • Our primary aim is to study the following question: given a function f dened on a subset of the real line R, and a point x0, describe the behavior of the sequence x0, f(x0), f(f(x0)), f(f(f(x0))), .... • Notation: Since we will frequently speak of the iterate of a function, we dene f n(x) to be the result of iterating f a total of n times on x. Thus, f 1(x) = f(x), f 2(x) = f(f(x)), f 3(x) = f(f(f(x))), and in general, f n(x) = f(f n−1(x)) for any n ≥ 2. We also adopt the convention that f 0(x) = x, the result of applying f zero times. ◦ This conicts with the convention, often used elsewhere, that the expression sin2(x) is to be interpreted as [sin(x)]2. In these notes we will therefore avoid the iterated function notation with explicitly-written trigonometric functions, and write explicitly that a function is being squared (when such a thing occurs). ◦ Do not confuse the iterated function notation with the notation for higher-order derivatives: f 3(x) means the triple iterate f(f(f(x))), while f (3)(x) means the third derivative f 000(x). 1.2.1 Denitions and Examples n • Denition: The orbit of x0 under f is the sequence x0, x1, x2, x3, ... where xn = f (x0). The value x0 is called the seed or initial point of the orbit. 1 • Example: Describe the orbits of x = 2, 0, 1, and under the function f(x) = x2. 0 2 ◦ For x0 = 2, the orbit is 2, 4, 16, 256, 65536, and so forth. This orbit clearly grows to 1. ◦ For x0 = 0, the orbit is 0, 0, 0, 0, 0, 0, and so forth. This orbit remains xed at 0. ◦ For x0 = 1, the orbit is 1, 1, 1, 1, 1, 1, and so forth.