It is often useful to think of T: R R, T(x) = Ax, as a time...

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It is often useful to think of T: R" R", T(x) = Ax, as a "time step map" for an evolving system in the following way: if x is the measurement of some quantity "now". then T(x) is what the measurement of that quantity would be "1-unit of time in the future". The composition T" (x) would then mean the measurement n-units of time in the future. We call this a discrete dynamical system (DDS). Consider the three discrete dynamical systems, T, T2, and T3 [0.85 0.05] 0.05 0.85 associated to: 1.1 0 [0.98 -0.1] A 0 0.9 A2 0.1 0.98 A3 = (a) GeoGebra is excellent for visualizing discrete dynamical systems of the plane. Suppose the initial measurement is P = (1, 1). Define a slider n with values 0 through 100 and step 1. Let Q=A"P. Turn on the trace of Q. As you slowly increase n from 0 to 100, you will visually capture the evolution of the system if you started at P. Save and include image for each of the three above systems. (After you do this, play with other starting values P to see how the initial measurement affects long term behavior.) (b) In complete sentences, verbally describe the progression P makes. Does it seem to limit towards a point? In what way does it move? (c) For each DDS, come up with a phenomenon that could be modeled by it. Explain. It is often useful to think of T: R" R", T(x) = Ax, as a "time step map" for an evolving system in the following way: if x is the measurement of some quantity "now". then T(x) is what the measurement of that quantity would be "1-unit of time in the future". The composition T" (x) would then mean the measurement n-units of time in the future. We call this a discrete dynamical system (DDS). Consider the three discrete dynamical systems, T, T2, and T3 [0.85 0.05] 0.05 0.85 associated to: 1.1 0 [0.98 -0.1] A 0 0.9 A2 0.1 0.98 A3 = (a) GeoGebra is excellent for visualizing discrete dynamical systems of the plane. Suppose the initial measurement is P = (1, 1). Define a slider n with values 0 through 100 and step 1. Let Q=A"P. Turn on the trace of Q. As you slowly increase n from 0 to 100, you will visually capture the evolution of the system if you started at P. Save and include image for each of the three above systems. (After you do this, play with other starting values P to see how the initial measurement affects long term behavior.) (b) In complete sentences, verbally describe the progression P makes. Does it seem to limit towards a point? In what way does it move? (c) For each DDS, come up with a phenomenon that could be modeled by it. Explain.

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