notes01_a - WEEK #1: Discrete-Time Dynamical Systems Goals:...

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Unformatted text preview: WEEK #1: Discrete-Time Dynamical Systems Goals: Introduce Discrete-Time Dynamical Systems. Find simple analytic solutions. Generate graphical solutions through cobwebbing. Textbook reading for Week #1: Read Sections 1 . 1 1 . 5 2 Modeling 101 To prepare for this topic, you should read Section 1.5 in the textbook. Mathematics and Biology can be combined to create powerful models of phenomena we see in the world around us, and to let us predict the consequences of changes. A powerful form of model is the Discrete-Time Dynamical System . We will be exploring this type of model for the next three weeks, and will return to it several more times during the year. Example: Consider an experiment where salt crystals are grown in a super-saturated solution. Crystals of diFerent initial sizes are put in the solution, grown for a day, and then the size of the new crystal is measured. It produces the following table: Week 1 Discrete-Time Dynamical Systems 3 Initial Mass (g) Mass 1 day later (g) 10.4 15.6 4.3 6.45 7.9 11.85 12.2 18.3 14.1 21.15 5.7 8.55 Represent the relationship between the starting and later mass as a function, using a graph Frst and a formula if you can Fnd one. 4 Initial Mass (g) Mass 1 day later (g) 10.4 15.6 4.3 6.45 7.9 11.85 12.2 18.3 14.1 21.15 5.7 8.55 Week 1 Discrete-Time Dynamical Systems 5 Updating functions Functions which map the same measurement at one time point to another time point (i.e. before and after measurements) are called updating functions . They are usually represented in the form m t +1 = f ( m t ) 6...
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notes01_a - WEEK #1: Discrete-Time Dynamical Systems Goals:...

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