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).