lab12.desc

lab12.desc - LAB#12 Linearization Goal Investigate the...

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Linearization Goal : Investigate the local behavior of a nonlinear system of diﬀerential equations near its equilibrium points by linearizing the system. Required tools : Matlab routine pplane ; eigenvalues and eigenvectors. Discussion In the last lab (Lab # 11) you classiﬁed equilibrium points in a linear system of equations as sources, sinks, centers and saddles. (Recall, an equilibrium point is a sink if all solutions which begin suﬃciently close to it converge to it; it is a source if all solutions suﬃciently close to it move away from it; it is a center if all solutions which begin suﬃciently close to it “loop around” it, i.e. they return to their initial position after a ﬁnite amount of time; the equilibrium point is a saddle if some solutions converge to it and some move away from it.) You also saw that for a linear system, the eigenvalues of the corresponding matrix determine what kind of equilibrium point the origin must be. In this lab, we investigate how to determine the nature of the equilibrium points of a nonlinear system. Assignment (1) Consider ﬁrst the linear system ± x 0 = - 3 x +( 2) y y 0 =( 2) x - 2 y ( * ) (a) Use pplane with window | x |≤ 10 and | y |≤ 10 to plot several orbits for the system ( * ). What kind of equilibrium point does the origin (0 , 0) seem to be? Print out your plot. (b) In the pplane Options ” menu, select “ Erase all solutions ” and use pplane with window | x |≤ 0 . 1and | y |≤ 0 . 1 to plot several orbits for ( * ). Print out this plot. Notice that all of the orbits seem to be tangent to one particular line at the origin. (c) Let A be the 2 × 2 matrix for the system ( * ). As in Lab #11, use the com- mand >> [B,D]=eig(A) to ﬁnd the eigenvalues and corresponding eigenvectors for A . Use this infor- mation to prove that the behavior you observed in (a) is correct. How does the line noted in (b) relate to the eigenvectors of A ? (2)

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lab12.desc - LAB#12 Linearization Goal Investigate the...

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