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# psheet2 - in the above equation and plot the graph of the...

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Data Analysis Modelling and Simulation MATH1711 Epiphany Term Practical Sheet 2 Aim: To calculate sequences of numbers from (nonlinear) re- currence relations Note before you start the second sheet ﬁnish the ﬁrst . You may ﬁnd it useful to know that To insert a new line put the cursor at the start of the line and use the menu option insert | Maple input To see the output from a calculation replace the : by a ; Find your problems (and help) on http://fife.dur.ac.uk:8080/AiM/ . The “ﬂower” example with various parameters The purpose of this example is to remind ourselves that recurrence relations generate sequences of well-deﬁned numbers. All of these examples are “second-order” and are the same/similar to examples analysed in lectures. Adding understanding to problem 5(b) Next the sequence generated by the ﬁrst-order diﬀerence equation is analyzed y n +1 = ay n + x n , where x n = δ n and y 0 = 0 . Some interesting values for a are 0.5, - 0.5, 1.0, - 1.0, 1.5 and - 1.5.

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Linear recurrence relations in signal processing Set x n = cos( 100 )+ ( - 1) n 10
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Unformatted text preview: in the above equation and plot the graph of the sequence x n and y n against n (good values are n = 200 and a = 0 . 5 ,-. 5). Do you nd comparable behaviour for other values for a ? Malthus population model, a particular choice of pa-rameters Changing an apparently harmless parameter can have a dramatic eect on the behaviour of a nonlinear recurrence relation. Investi-gate the nonlinear recurrence relation y n +1 = ay n (1-y n ) where y (0 , 1) . Notice that if 0 &amp;lt; a &amp;lt; 4 then y n (0 , 1). There are three cases to consider (what is dierent about each?) (a) 0 &amp;lt; a &amp;lt; 1 (try a = 0 . 5); (b) 1 &amp;lt; a &amp;lt; 3 (try a = 1 . 5 , 2 . 5); (c) 3 &amp;lt; a &amp;lt; 4 (try a = 3 . 2 , 3 . 52 , 3 . 58 , 3 . 83.) Try taking lots of dierent starting values y (0 , 1). In the last case, (c), on certain occasions you may want to plot a histogram. To do this type with(stats): with(stats[statplots]): histogram([y(n) \$n=0. .100],area=count,numbars=50) ;...
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psheet2 - in the above equation and plot the graph of the...

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