Section I: A Simple Oscillating Model
Oscillation is the most common form of dynamic behavior in the world. As a consequence, it is important for a system dynamicist to intuitively understand its cause. Moreover, it is important for him or her to be able to translate an oscillating system from one of its traditional mathematical forms to a system dynamics model, so that it can be analyzed from the system dynamics perspective.
1) Read Clarence Peterson's Chicago Tribune article (Clarence Petersen Article.pdf): “As Usual, Boy + Girl = Confusion.” Notice that Mr. Peterson is having trouble intuitively relating the model’s structure (i.e., the equations) to its purported behavior (i.e., a “never-ending cycle of love and hate”). Indeed, he claims to have appealed to his father for help.
2) Translate Strogatz’s model into a system dynamics model. Give intuitively meaningful names to the stocks and flows. For example, you might change dr/dt to something like: “ChangeRomeoLove”.
3) Select initial values for the model's stocks. Note that Peterson does not say what these values should be. However, he does provide a clue by indicating that “positive values r, j signify love; negative values signify hate.” What happens if you select zero for the initial values? What would be an intuitive interpretation of a stock with a value of zero?
4) Specify the model's parameters. Again, Peterson does not explicitly mention what they should be, but does indicate that they must be positive “to be consistent with the story.” This clue may be a bit misleading, however, as the parameter “a” has a negative sign in front of it. In other words, when “a” is specified as a positive number, the minus sign makes it a negative parameter. What happens if you make both “a” and “b” zero? Be sure to specify what the intuitive meaning of the parameters is. Hint -- be sure that the model is dimensionally correct.
5) Simulate the model using Euler's method, your initial choice for the parameters and initial values, and a step size of .5. Plot the two stocks and the number zero on the same time series graph (specify “zero” as a constant with a value of 0, connected to nothing). Be sure all three variables are plotted on the same scale. In addition, plot the two stocks against one another on a second graph to create the model's phase plane. A linear harmonic oscillator, such as the one you are working with, generates a sustained oscillation with constant amplitude and periodicity. On the phase plane, it generates a corresponding closed loop or circle. If you see a spiral in the phase plane and a corresponding “exploding oscillation” on the time series plot, it is due to integration error and not to the model's structure. To clear up the integration error, try cutting dt in half and re-simulating. Does it help? If it does, but the model still generates a spiral and exploding oscillation, keep cutting dt in half and re-simulating until the correct behavior appears. Notice the change in time it takes to simulate the model each time you cut dt in half. Why you think this happens? Do you think that it is a good idea to always cut dt in half when examining the behavior of a new model for the first time? Why?
6) Revert back to your first model run (i.e., a dt of .5 and your original parameters and initial values). Simulate the model using a second order Runge Kutta method. How does this run compare to your first one? What’s going on? Cut dt to .1 and re-simulate. Does the model's output look the way it should? How long does it take to simulate the model with a second order Runge Kutta method and a step size of .1, vis-à-vis the time it took using Euler's method with its appropriate step size?
7) Retain the second order Runge Kutta method and your original parameters and initial values. Increase the values of “a” and “b” and re-simulate. What happens to the periodicity of the cycle? Does it take more or less time for the model to complete one cycle? Increase “a” and “b” again and re-simulate. What happens to the periodicity of the cycle this time? Can you discern a relationship between the model’s parameter values and the periodicity of its cycle? For a given dt (say .5) and integration method (say Euler’s), do larger or smaller values of “a” and “b” yield more integration error? Why?
8) Retain the second order Runge Kutta method and your original parameters and initial values. Increase the stock-s initial values and re-simulate. What happens to the amplitude of the cycle, relative to your first Runge Kutta simulation? Boost the initial values again and re-simulate. What happens to the amplitude now? Again, can you discern a relationship between the model’s initial values and the amplitude of its cycle? Speculate -- if you were to make a general statement about all linear systems that oscillate, what factors would you say determine the character of their cycles?
9) Peterson claims that Romeo and Juliet are kept apart, not by their families, but by “Romeo's fickleness.” Provide a verbal description of the cycle and Romeo’s fickleness. That is, talk your way through the “never-ending cycle of love and hate.”
10) Peterson also notes that: “The news is not all bad. According to the equation...Romeo and Juliet „manage to achieve simultaneous love one quarter of the time.‟” Use the phase plot to show that this is true. Hint: You may wish to draw a couple of lines on the phase plot. Explain what you have done.
11) Draw a causal loop diagram of the model. What kind of loops are involved? Is the model consistent with the system dynamics heuristic for oscillation? Where’s the delay?
12) Analyze the model in terms of a tempestuous relationship between a man and a woman. Is it a “good” model of such a relationship? Why or why not?
13) In the last paragraph of the article, Peterson claims that his father's advice was to: “Substitute the Hatfields and McCoys...”, so that he (Peterson) could better understand the model. Do you think this is good advice? Draw a causal loop diagram of a feud. Is it essentially the same as, or different from, the one you drew for Romeo and Juliet?
Section II: A More Complex Oscillating Model
If one pushes past Peterson’s Chicago Tribune article and examines Steven Strogatz’s original piece (Strogatz - Love Affairs and Differential Equations.pdf), one finds that he offers a second, more general, model of dyadic relationships:
dr/dt = a11*r + a12*j
dj/dt = a21*r + a22*j
where: r(t) = Romeo’s love/hate for Juliet at time t
j(t) = Juliet’s love/hate for Romeo at time t [1]
According to Strogatz, much of the fun in analyzing [1] comes from the specification of its parameters. That is, the parameters aik (i,k = 1,2) can be either positive or negative, and their signs determine the “romantic style” of each participant. He thus claims that (a11, a12 > 0) would “characterize an „eager-beaver‟” or someone stimulated by both his/her partner’s love and his/her own affectionate feelings, and (a21 > 0, a22 < 0) would characterize a “cautious lover” or someone excited by his/her partner’s love but frightened by his/her own feelings. As an exercise, Strogatz recommends naming the two other romantic styles (i.e., a11, a12 < 0; and a21 < 0, a22 > 0), and providing “romantic forecasts” for various pairings of styles. Indeed, in terms of the latter task, he poses the question of whether “a cautious lover...[can] find true love with an eager-beaver.”
1) Translate [1] into a system dynamics model and parameterize it for the eager-beaver/cautious lover case. Draw a causal loop diagram for the model. What are the polarities of the feedback loops involved? Determine which parameters control the strength of the loops. Can you see how the issue of loop polarity is intertwined with the issue of parameter selection and hence with the specification of romantic styles?
2) Simulate the model with each parameter having the same magnitude (but not necessarily the same sign). Is it possible for a cautious lover to find true happiness with an eager-beaver? What is the behavior of the model? What is the dominant form of feedback (positive or negative)?
3) Give names and intuitive descriptions to the two other romantic styles possible in the model. Select a combination of two styles and simulate. What happens? Provide an intuitive description (i.e., in terms of a romantic relationship) and a more technical “feedback loop description.”
4) Try other combinations of romantic styles. Be sure to provide an intuitive description and a more technical “feedback loop description.” of each result. If you come across a combination that yields an exploding oscillation (and its not due to integration error -- remember to cut dt in half as a test), determine where it’s coming from. Since actual systems can’t yield exploding oscillations (at least not for long), what might be missing from the model? Hint: Remember, you’re dealing with a linear model.
5) Critique [1] as a model of romantic relationships.
Hint: For help on this homework read: Radzicki, Michael J. 1993. "Dyadic Processes, Tempestuous Relationships and System Dynamics." System Dynamics Review 9(1): 79-94.
Please present your answers to the questions in this homework assignment in a MS Word or PDF document