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Unformatted text preview: MA 2930, Feb 23, 2011 Worksheet 5 1. Stability of critical points For each of the following autonomous equations find its critical points, deter- mine if they are stable/unstable/semi-stable, and draw the phase line as well as qualitative sketches of representative solutions. (1) dy/dt = y 2 ( y- 1)( y- 2) (2) dy/dt = (1 + y 2 ) arctan y (3) dy/dt = e y 2- 2 (4) dy/dt = sin 2 y (1) Critical points are the values of y for which dy/dt = 0. Clearly they are y = 0 , 1 and 2 in this case. To determine their stability we can either look at the sign of dy/dt on either side of them, or calculate f ( y ) and look at its sign at the critical point. Here the first method is easy to apply. If y < 0 (say y =- 1), y > 0. If 0 < y < 1 (say y = 0 . 5), y > 0 as well. So y = 0 is a semi-stable critical point: from below it attracts, but from above it repels. If 1 < y < 2, y < 0. So y = 1 is stable. It attracts the flow from either side. If y > 2, y > 0. So y = 2 is unstable. The system flows away from it on either side. I’ll let you draw the phase line etc. because I don’t have a program for drawing pictures! (2) Critical points are where (1+ y 2 ) arctan y = 0, i.e., where arctan y = 0, i.e., y = 0....
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