Solved: In each of 1 through 3, construct a suitable Liapunov function of the form ax2 +

Chapter 9, Problem 1

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QUESTION:

In each of 1 through 3, construct a suitable Liapunov function of the form ax2 + cy2, where a and c are to be determined. Then show that the critical point at the origin is of the indicated type. dx dt = x3+xy2, dy dt = 2x2 yy3; asymptotically stable

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QUESTION:

In each of 1 through 3, construct a suitable Liapunov function of the form ax2 + cy2, where a and c are to be determined. Then show that the critical point at the origin is of the indicated type. dx dt = x3+xy2, dy dt = 2x2 yy3; asymptotically stable

ANSWER:

Step 1 of 3

We have .

We note that .

We want to check that  is an isolated critical point Suppose  and . Then we must have  and so . Then we must have  which is impossible. Hence  is the only critical point.

 

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