Dynamical Systems: Scope and Sequence

Introduction

Dynamical systems provide an accessible way of introducing the subject of change to students at many levels and are an excellent source of applications for making mathematics more meaningful for students. These applications include models of population growth and decline, predator-prey relationships, traffic flow problems, battles, river meanders, weather, heartbeat rates, simulation of landscapes and flora, and so on. The goal of the study of dynamical systems is to create and analyze models of systems in such a way that questions about the system can be asked and answered.

The subject has recently become accessible for several reasons. The increase in computing power makes numerical techniques easy to use. Iteration is a key technique for the study of dynamical systems and is easily performed with calculators or computer software. Students no longer need to use the standard techniques of the calculus of differential equations. The improvement in visual imagery has made it easier to graphically analyze data and to display images of dynamical systems. The display may be a single snapshot or a string of pictures presented as a slide show or a short film.

A variety of approaches can be taken with students at many levels. Students can begin to analyze data and express relationships verbally. (System dynamics focuses on relationships rather than on the parts.) Students will develop concepts and images for fast and slow change, increasing, decreasing, and steady-state systems. Numerical analysis can be made by finding differences, ratios, and patterns. Data can easily be represented graphically; first using time series graphs and then by curve fitting or regression models. Dynamical analysis leads to the development of discrete models based on difference equations. Advanced students can set up and analyze differential equations using numerical techniques like Euler's Method or the analytic techniques of traditional calculus classes. Some numerical techniques like Euler's Method are accessible to algebra students.

The remainder of this paper is a brief summary of some of the major concepts and techniques discussed during the institute. The organization is arbitrary: many ideas can be introduced over a wide range of skill levels. Teachers should not hesitate to extend "simple" activities for more advanced students or to simplify more difficult activities for beginning students. At the end of each major heading is a list of the activities contained in this module that should be appropriate for that topic. However, many of these activities have broad application and the placement of activities should not be considered immutable. The numbers correspond to the list of activities attached to this paper.

Introducing the Student to Dynamical Systems

The basic concepts of and techniques applied to the study of dynamical systems could be introduced to young children, and certainly are appropriate for middle school students. The key concept underlying the numerical techniques is iteration. To "iterate" an equation or function means to evaluate it repeatedly. Usually this is done recursively which means that the output of one iteration is used as the input for the next iteration. Recursion is the key element of feedback loops, a basic element of system dynamics. Feedback loops can be either positive or negative depending on whether the effect on the output of the function is to increase the output or to decrease the output. The value which is first used as input is called the "seed", and the sequence of output values is called the "orbit" of the seed. These basics can be explored easily on a calculator or with a spreadsheet.

Students can begin their analysis of the data that they collect for their dynamical systems by applying some basic tools of data analysis to tables of information. They can calculate first and second order finite differences and look for patterns. They can plot points, putting the independent quantity on one axis and the dependent quantity on the other axis. Students can then give a verbal description of the relationships that they observe and make tentative predictions about the future behavior of the system based on the model. Making tables and plotting points can be done by hand or with a spreadsheet on a computer or on some calculators. Many old problems, such as the Handshake problem, and many new problems provide an opportunity for the student to create tables of data, analyze them, and make predictions. Students with algebra experience can derive the recursive formulas for the data, and those familiar with matrices can derive the closed-form of the model.

Geometric approaches can also be used to introduce the student to the basic concepts of dynamical systems. Geometric figures can be iterated just like numbers or expressions. Cantor Sets, the Koch Snowflake, and the Sierpinski Triangle can be created by recursively iterating a seed figure. The Snowflake and Triangle exhibit "self-similarity" which means that pieces of the figure are copies of the whole. Figures which exhibit self-similarity are called "fractals". Comparisons of the scale factor for consecutive iterations and the number of pieces used can be made to determine the fractal dimension. The Chaos Game provides an opportunity to introduce the concept of order resulting from a random process.

Activities: 4, 9, 11, 15, 19, 23, 30, 31, 33

Linear Dynamical Systems

Linear functions grow naturally from population models and financial models. Tables of data are used to derive difference equations of the form M(k+1) = aM(k) + b. (More advanced students can use regression to model the data with linear functions.) These equations can be derived recursively. These techniques can be used to develop forms for loans, savings, lotteries, and a general financial model; for population models; for radioactive decay; and for many other problem situations. By varying the value of "a", the analysis of linear systems can be used to introduce "fixed points"-both attracting and repelling. These points occur where the graph of the function crosses the line y = x. The character of the fixed point depends on the slope of the line. Stability or lack of it is important in analyzing these functions. Using spreadsheets, calculators, or software such as "Stella" or "MathCad", will allow the students to explore the effect of small changes in the seed for different values of "a". The trigonometry problem which requires the student to find the angles for tunneling from both sides of a mountain is an example of a problems that can be used to explore stability. More advanced students can work with problems for which "a" and "b" are matrices or complex numbers.

The data can be analyzed by fitting a linear curve to the data. Writing an equation or function to fit the data is one part of the mathematical modeling process. The goal is to find a function which accurately reflects the data and to use that function to answer questions about the dynamical system. The domain of the problem situation is an important consideration, and the values of "m" (the slope) and "b" (the y-intercept) can be interpreted in terms of the problem context. The "residuals" are the differences between the actual data values and the predicted values. Residuals are a measure of the closeness of the fit. Residuals can be entered into a table and/or graphed against the independent variable for analysis. The sum of the absolute values of the squares of the residuals can also be used, and will provide the student with some background for the least-squares technique of curve fitting. Advanced students will see that iteration of a linear function leads to exponential growth.

Geometric analysis of linear dynamics can be done with Web Diagrams. These diagrams will provide a strong visual representation of the seed, the orbit, repelling fixed points, and attracting fixed points. They can be used to explore the effect of changing the slope at the fixed point. If the students are familiar with complex numbers, they can study the effects of different complex coefficients: rotations and dilations through the fixed points. Students who have worked with complex numbers, rotations, dilations, and translations can find equations to model the transformations which result in the Koch Snowflake or other self-similar curves.

Activities: 2, 4, 11, 12, 15, 19, 20, 23, 24, 29, 34

Quadratic Dynamical Systems

Quadratic dynamical systems result from more realistic modeling of populations and from many other applications. The logistic model, P(k+1) = P(k)*(1-g) - g*P(k)², is commonly used. There is a major difference between linear and quadratic dynamical systems. As the quadratic is translated vertically, the slope changes at the fixed point (where the curve intersects y = x) and, as a result, the character of the point changes. Remember that the character of the fixed point depends on the slope of the curve at that point. The character of points can be analyzed numerically with tables of values and graphically with Web Diagrams. The orbits of seeds may be fixed, periodic, chaotic, or they may escape. The critical points (derivative is zero) of the function are the points of interest. The basin of attraction of a fixed point can be illustrated on a number line. The "basin of attraction" is the set of seeds whose orbits reach the fixed point. A bifurcation diagram can be used to study the fate of the orbit of a seed as the parameter is changed. The ratio of values in the diagram leads to Feigenbaum's Number. Feigenbaum's Number is the ratio of consecutive bifurcation values. Population models based on the logistic equation provide an application on which to build these concepts. Students can use any of the following tools to analyze quadratic dynamical systems: spreadsheets, "Stella", "MathCad", calculators, or paper and pencil. Euler's Method can be used by both calculus and precalculus students to make approximations and recreate values for f(x) from the known values of f'(x).

Students can extend their data analysis skills with new techniques for analyzing the curve that they fit to the data. Similar looking sets of data may have greatly different functions, so the student needs to check the fit of the curve to the data carefully. If the data looks quadratic, graph (x, y). The resulting graph should be linear, if the data is quadratic. If the graph is not quite linear, the student can adjust the index of the radical to try to obtain a better linear fit. The concavity of the graph will indicate the direction and size of the needed adjustment. Students who are familiar with rational exponents may try a rational index. The reasoning behind the process uses the concept of the inverse of a function. If y = mx + b, then y = (mx + b)². This process can be generalized to finding the inverse of whatever type of function appears to fit the data. To test for an exponential fit to the data, graph (x, ln y). To test for a power fit to the data, graph (x, kth root of y) or (x^k, y). Even if you don't get the exact inverse, you still have the same type of function and the graph of the inverse should be linear. Spreadsheets, "Stella", "MathCad", calculators, and paper and pencil can be used to model and analyze quadratic dynamical systems.

Web diagrams can provide a geometric picture of the fixed points, the fate of orbits, and basins of attraction. Coloring each point on the number line according to the fate of that point's orbit will give the student a picture of the distribution. Use different colors for orbits that reach each of the fixed points, another color for orbits that are periodic, yet another for seeds whose orbits escape, and a last for seeds whose orbits are chaotic. Web diagrams and bifurcation diagrams can be produced by hand, on some calculators, and with programs for calculators and computers.

Activities: 1, 2, 3, 7, 10, 12, 16, 27, 28, 34, 35

Other Dynamical Systems

As students become familiar with the techniques of modeling dynamical systems, more complex systems and models can be introduced. Examples of such systems include wind chill data, models for minimizing the number of drug tests needed to determine who is using a particular drug, and determining for a taxi company where they should station their vehicles for optimum coverage of the city given the probabilities for moving from region to region in the city. Some population models including the Leslie Matrix Model have matrix coefficients. Students can study river meanders by plotting the distance downstream against the angle of deviation from the mean down valley direction. The result can be modeled with a sine curve. Such sine generated curves minimize change in direction, total work done in turning, and total erosion. Similar results appear in waving flags, stress lines in trees, and ocean currents. The basic processes for analyzing data and creating models are just the same as they are for the other dynamical systems.

Student models of these types of situations may include differential equations or other equations for which the standard analytic techniques of calculus may not help-there may be no analytic solution! Students can use Euler's Method to reconstruct values for the solution to differential equations that are difficult or impossible to solve. This technique permits investigation of a model without solving the associated differential equation. Since it is discrete, Euler's Method (as well as Newton's Method) may have periodic or chaotic behavior lurking behind the process. Sometimes a solution can be found for the differential equation, but it is not possible to isolate the dependent variable, students can substitute a value for the independent variable and graph the resulting equation to analyze the model. Different analyses may result from different choices of values for the independent variable.

Activities: 2, 3, 7, 8, 12, 16, 21, 27, 28, 35

Complex Quadratic Dynamical Systems

These systems have recently received considerable attention. The applications are just being developed: computer generated graphics, modeling of plant and animal forms, data compression, encryption, and many more. There are several reasons that these systems are receiving the attention that they are: they produce beautiful pictures and mathematics, it is a topic currently being researched, it is easily accessible through computer software, and it has an experimental nature. The goal of the study of these systems is to understand the fate of the orbit of a particular seed when we iterate a complex function. Some orbits escape (to infinity), others go to fixed points (immediately or eventually), others become periodic (immediately or eventually), and some become chaotic. The fates can be represented several ways: as the numerical sequence of the orbit, as a time series, as a histogram, as a web diagram, as a bifurcation diagram, as a Julia Set, or as a Mandelbrot Set. The Julia Set is a graph of all seed values whose orbits do not escape. The Mandelbrot Set is a graph of all the "c" values for which the orbit of zero does not escape. Note that zero is the critical point for a function of the form f(z) = z² + c. If the critical orbit does not escape for a particular Julia Set, then that Julia Set is connected. If the critical orbit does escape for a particular Julia Set, then the set explodes and there is chaotic behavior over the entire plane. The Julia Set then consists of infinitely many points and is called "fractal dust". (It is an example of a Cantor Set.) All Julia Sets are fractals, that is they are self-similar.

The Mandelbrot Set is not self-similar. Each point in the Mandelbrot Set is associated with a Julia Set which is connected. Each point outside the Mandelbrot Set is associated with a Julia Set which explodes into fractal dust. The Mandelbrot Set is reasonably well understood, and a "road map" has been developed which associates every rational and some irrational numbers with points on the boundary of the set. To make the map, each point in the plane is expressed in polar form with the angle expressed in turns rather than degrees or radians. According to DeMoivre's Theorem, when the complex seed is squared the angle is doubled. This doubling function or doubling map is a the key to understanding the Mandelbrot Set. When the measure of the angle's turn is expressed in binary form, the sequence of binary digits describes the orbit of that number. Repeating and eventually repeating binary numbers are associated with particular points on the Mandelbrot Set. The fate of irrational numbers (their binary sequence never repeats) is uncertain.

The unit circle can provide an easy introduction to most of this work. The orbits of points inside the circle go to zero, the orbits of points outside the circle escape, and the orbits of points on the circle are chaotic. These last points are the points of interest. Web diagrams also provide a picture of the process of tracing the fate of orbits. Computer software can be used to display and investigate the Mandelbrot Set and Julia Sets.

Activities: 6, 17, 26

Complex Dynamical Systems Other Than Quadratic

The fate of orbits for trigonometric and exponential functions can also be investigated. For Transcendental Functions, the Julia Set is the part where the chaotic behavior takes place (the part that is colored) rather than the stable part (usually colored black). Julia Sets are still self-similar, and the Mandelbrot Set shows up in these Julia Sets also. Those parameters for which Newton's Method fails in the complex plane are the Mandelbrot Set. Cubic functions are tougher because they have two critical points and therefore there are four complex numbers to work with-a four dimensional space.

Woodrow Wilson Leadership Program in Mathematics lpt@www.woodrow.org
The Woodrow Wilson National Fellowship Foundation webmaster@woodrow.org
CN 5281, Princeton NJ 08543-5281 Tel:(609)452-7007 Fax:(609)452-0066