ExercisesSolutions

# Classifying the fixed points using the eigenvalues 3

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Classifying the fixed points using the eigenvalues, ([3 , 7 , 11 . . . ] π/ 2 , 2 ) : bracketleftbigg 0 1 1 0 bracketrightbigg . Eigenvalues are 1 and - 1, and thus the fixed point(s) is a saddle point . ([1 , 5 , 9 . . . ] π/ 2 , 2 ) : bracketleftbigg 0 1 - 1 0 bracketrightbigg . Eigenvalues are I and - I , and thus the fixed point(s) is a center . (f) ˙ x = xy - 1 , ˙ y = x - y 3 The real fixed points are (1 , 1), ( - 1 , - 1) and ( I, - I ). The Jacobian of the system is bracketleftbigg y x 1 - 3 y 2 bracketrightbigg . Classifying the fixed points using the eigenvalues, (1 , 1) : bracketleftbigg 1 1 1 - 3 bracketrightbigg . Eigenvalues are - 3 . 2361 and 1 . 2361, and thus (1 , 1) is a saddle point . ( - 1 , - 1) : bracketleftbigg - 1 - 1 1 - 3 bracketrightbigg . Eigenvalues are - 2 and - 2, and thus ( - 1 , - 1) is a stable node . 14

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3. Purpose: To compare computer generated phase portratis with initial sketches. Exercise: For each of the nonlinear systems in Exercise 2, plot a computer-generated phase portrait and compare to your approximate sketch. Solution: The required computer-generated phase portrait for each system was created in Maple and are shown below. –4 –2 0 2 4 y –4 –2 2 4 x Figure 12: Phase portrait for Exercise 3a –8 –6 –4 –2 0 2 4 6 8 y –2 –1 1 2 x Figure 13: Phase portrait for Exercise 3b 15
–4 –2 0 2 4 y –4 –2 2 4 x Figure 14: Phase portrait for Exercise 3c –4 –2 0 2 4 y –4 –2 2 4 x Figure 15: Phase portrait for Exercise 3d –4 –2 0 2 4 y –4 –2 2 4 x Figure 16: Phase portrait for Exercise 3e 16

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–4 –2 0 2 4 y –4 –2 2 4 x Figure 17: Phase portrait for Exercise 3f 17
4. Purpose: To use qualitative arguments to deduce the phase portrait of a system. From [Str94], Problem 6.6.6 Exercise: Consider the reversible system ˙ x = y (1 - x 2 ) , ˙ y = 1 - y 2 . (a) Plot the nullclines ˙ x = 0 and ˙ y = 0. (b) Find the sign of ˙ x, ˙ y in different regions of the plane. (c) Calculate the eigenvalues and eigenvectors of the saddle points at ( - 1 , ± 1). (d) Consider the unstable manifold of ( - 1 , - 1). By making an argument about the signs of ˙ x, ˙ y , prove that this unstable manifold intersects the negative x -axis. Then use reversibility to prove the existence of a heteroclini trajectory connecting ( - 1 , - 1) to ( - 1 , 1). (e) Using similar arguments, prove that another heteroclinis trajectory exists, and sketch several other trajectories to fill in thephase portrait. Solution: (a) The nullcline plot of ˙ x = 0 and ˙ y = 0 can be seen in Figure 18. –2 –1 0 1 2 y –2 –1 1 2 x Figure 18: Nullcline plot for Exercise 4a (b) Figure 19 shows the signs of ˙ x, ˙ y in the different regions of the plane. dy/dt<0 dx/dt>0 dy/dt>0 dx/dt>0 dy/dt>0 dx/dt<0 dy/dt<0 dx/dt<0 dy/dt<0 dx/dt<0 dy/dt>0 dx/dt<0 dy/dt>0 dx/dt>0 dy/dt<0 dx/dt>0 dy/dt<0 dx/dt>0 dy/dt>0 dx/dt>0 dy/dt>0 dx/dt<0 dy/dt<0 dx/dt<0 –2 –1 0 1 2 y –2 –1 1 2 x Figure 19: The sign of ˙ x, ˙ y in different regions of the plane.

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