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MATH 616-010 Modeling in Applied MathematicsProf. D. A. Edwards Sept. 13, 2018
Liapunov Functions
For the case wherex = �y � x
3
y = x � y3
(1)
we found that a Liapunov function was given by
V = x2 + y
2,
and that the origin is stable.
-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
Phase plane of (1). Note the slow approach to the origin.
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MATH616Liap.2
For the case wherex = �y + 3x
3
y = 2x + y3
(2)
we found that a Liapunov function was given by
V = 2x2 + y
2,
and that the origin is unstable.
-2.5 0 2.5
Phase plane of (2).
MATH616Liap.3
For the case wherex = �y � x(x � 1)
y = (2x � 1)[x(x � 1) � y](3)
we found that a Liapunov function was given by
V = x2(x � 1)2 + y
2.
Here the origin is unstable and (1,0) is stable.
0 2.5
Phase plane of (3).