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Unformatted text preview: 640:244 FALL 2009 Lab 5: A Nonlinear System This Maple lab is based in part on earlier versions prepared by Professors R. Falk and R. Bumby of the Rutgers Mathematics department. Introduction. In this lab we use Maple to study the phase plane of an autonomous nonlinear system of two differential equations, i.e., a system of the form x ′ ( t ) = F ( x, y ) , y ′ ( t ) = G ( x, y ) . (1) The distinguishing feature of an autonomous system is that the expressions defining the functions F and G do not contain the independent variable t . This allows many properties of the solutions to be studied using the curves, called trajectories , that show the path in the xy plane followed by the solutions. (It is an easy exercise to show that if an initial condition is on a trajectory, then the whole solution follows that trajectory). The Maple command DEplot may be used to draw trajectories and direction fields for such systems. Please obtain the seed file from the web page and save it in your directory on eden, then prepare the Maple lab according to the instructions and hints in the introduction to Lab 0. Turn in only the printout of your Maple worksheet, after removing any extraneous material and any errors you have made. 0. Setup. As usual, the seed file begins with commands which load the required Maple packages: with(plots): , with(DEtools): , and with(LinearAlgebra): . 1. An autonomous nonlinear system: a. The equation and its critical points. We will study the nonlinear system x ′ = 4 x + xy x 2 = x (4 + y x ) y ′ = 6 x + 6 y xy x 2 = (6 x )( x + y ) . (2) The equilibrium solutions are [ x = 0 , y = 0], [ x = 2 , y = 2], and [ x = 6 , y = 2]. Maple can obtain these by using the instructions F:=2*y  2*x + x*y  x^2; G:=4*y + 4*x  x*y  x^2; eqpts:=solve(F,G,x,y); which are included in the seed file. Also included are instructions to define the differential equations: dex:=diff(x(t),t)=eval(F,{x=x(t),y=y(t)}); dey:=diff(y(t),t)=eval(G,{x=x(t),y=y(t)}); For later convenience we have defined F and G to depend on the variables x and y , but in the differential equations we must write these variables as x ( t ) and y ( t ); the eval command makes this substitution. b. The direction field and nullclines. The next instructions establish an appropriate range for the independent variable and a useful viewing window, and construct a plot of the direction field of this system. The plot consists of small arrows pointing the way of the trajectories in the square 4 ≤ x ≤ 8, 6 ≤ y ≤ 6. trange := 8..8: window:=x=4..8,y=6..6: df:=DEplot([dex,dey],[x(t),y(t)], trange, window,color=GREEN): 1 640:244 FALL 2009 (Note the colon at the end of the instruction defining the plot, which suppresses output of the plot structure). The color option is used to give a better view of the nullclines and solution curves to...
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 Fall '09
 Ming
 Linear system, Nonlinear system, phase plane

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