notes02_a - WEEK #2: Cobwebbing, Equilibria, Exponentials...

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WEEK #2: Cobwebbing, Equilibria, Exponentials and Logarithms Goals: Analyze Discrete-Time Dynamical Systems Logs and Exponentials Textbook reading for Week #2: Read Sections 1 . 6 1 . 7
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2 Graphical Analysis To prepare for this topic, you should read Section 1.6 in the textbook. A powerful way to understand the behaviour of discrete-time dynamical systems is to use graphs to understand the behaviour. Example: The Frst 4 steps of the solution to the morphine system, a t +4 = 0 . 7 a t + 10 . are shown below, for the starting point a 0 = 0 mg, and a 0 = 50 mg.
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Week 2 – Cobwebbing, Equilibria, Exponentials and Logarithms 3 First, we draw the solution graph , which is usually what we want in the end. This is the prediction of morphine levels over time . t a t a t +4 t a t a t +4 0 0 10 0 50 45 4 10 17 4 45 41.5 8 17 21.9 8 41.5 39.05 12 21.9 25.33 12 39.05 37.335
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4 For the same example, however, it is useful to see how we can also view the morphine levels over time in the updating function , a t +4 = 0 . 7 a t + 10 . This process is called cobwebbing and uses the “ y = x ” line as an important reference line.
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Week 2 – Cobwebbing, Equilibria, Exponentials and Logarithms 5 t a t a t +4 t a t a t +4 0 0 10 0 50 45 4 10 17 4 45 41.5 8 17 21.9 8 41.5 39.05 12 21.9 25.33 12 39.05 37.335
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6 Association: why is the y = x line important when drawing inverse functions ? Why is the y = x line, or a t = a t +4 line in this example, important on the graph of the updating function?
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Week 2 – Cobwebbing, Equilibria, Exponentials and Logarithms 7 When the updating function a t +4 = 0 . 7 a t + 10 crosses the a t = a t +4 line, what is special about that a value? ConFrm this by Fnding a t +4 if we start at the intersection value, a t = 100 3 33 . 333.
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8 Cobwebbing Cobwebbing is a graphical technique for visualizing the solution to a discrete-time dynamical model using only the updating function . From Section 1.6, 1. Graph the updating function and the diagonal. 2. Starting from the initial condition on the horizontal axis, go “ up to the updating function and over to the diagonal ”. 3. Repeat until you have taken the speci±ed number of steps, or until you ±nd the pattern. 4. Sketch the solutions at time 0, 1, 2, and so forth on an m t vs. t solution graph.
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Week 2 – Cobwebbing, Equilibria, Exponentials and Logarithms 9 Example: Logistic Curve We will now use cobwebbing to predict the long-term behaviour of the solution, given the graph of the updating function below. We will study the eFect of three diFerent starting points. 0 10 20 30 40 50 0 10 20 30 40 50 m t m t + 1
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Generate a cobwebbing diagram on the updating function graph, and then show the approximate solution for the system, given a starting point of m 0 = 50. Note that we have zoomed in on the graph for clarity. t (steps)
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This note was uploaded on 12/19/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Fall '08 term at University of Ottawa.

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notes02_a - WEEK #2: Cobwebbing, Equilibria, Exponentials...

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