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 Dynamic Feedback System Models (c) 2009, Ed Seykota   Tutorial on Dynamic Feedback Models: Simple Level and Rate Model We represent  model structure with a Levels-and-Rates Diagram We further define the structure with specific equations.   In this case, In steps up from 0 to 1 at T = 20 and from 1 to 2 at T = 30. Out steps up from 0 to 1 at T = 20 and from 1 to 2 at T = 30. We simulate the model to see how it behaves.   In this case, At T = 0, In and Out are both zero so Level stays constant. At T = 10, In increases to 1 so Level ramps up from 0 to 10. At T = 20, Out increases from 0 to 1 so Level stays constant. At T = 30, Out increases from 1 to 2 so Level ramps down to 0. At T = 40, In increases from 1 to 2 so Level stays constant. First-Order Feedback "First Order" Indicates the number of Levels = 1. Feedback indicates information from Level feeds back to control Rate.   In this case, Rate = Level / Time_Constant For TC = -5 Months, we observe negative feedback, in which Level drops to 1/e of its original value every 5 months.   This is a basic pattern that occurs throughout nature and ecomonics. For TC = +5Months, we observe positive feedback, in which Level increases to e times its original value every 5 months.     This pattern occurs for some successful businessmen. Pendulum Model The Structure The Equations The Simulation   In this case, the pendulum starts off from the "bottom" position with high velocity. The pendulum "wraps around" a couple times and then settles down into oscillatory mode with gradual decay.     Notes: The period varies with amplitude in this model and in real-world pendulums.   The amplitude-specific property of period does not generally appear in high school and physics texts, since solving for this effect with the differential calculus is a formidable task.   The classic solution for the period P = 2 * pi * sqrt(length/Gravity_Acceleration) gives P = 2 * 3.14 * sqrt(10/9.8) P ~ 6-1/2   We see this as the approximate period in the model. Simulation Showing Angle and G-Acceleration Kondratieff Wave Model We can use dynamic feedback modeling to investigate economics.   This model shows one formulation of the Kondratieff Wave theory about the capital expansion-contraction cycle.   The period is, according to Kondratieff about 60 years. This matches real world observations and this model.   This model shows periods of intense capital formation and long periods of capital depreciation in which out-of-date factories disappear, making room for new ones during the next expansion cycle.     Model Structure Try a Stimulus Package in the Model In this case, we add a form of capital stimulus package.   The result of this particular form is that the cycle time lengthens, and the amount of capital remains generally higher.   The stimulus package, in effect, prevents the system from purging old capital and acquiring new capital.   Dynamic Feedback System Models provide a systematic way to define and test assumptions, see how behavior flows from assumptions, learn how systems operate and investigate sensitivity to various policies.