for the system x x 3 2y 3 y x 3 (1) prove that V (x,y) ax 2 by 2 cannot be used as a Lyapunov function to show stability of the origin for any a and b Hint consider points x 2 , y for small (2) what are values of a and b such that V (x,y) ax 4 by 4 is a Lyapunov function (3) conclude about the stability of the origin from the Lyapunov function in (2)

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Question: for the system x = x 3 + 2y 3 y = x 3 (1) prove that V (x,y) = ax 2 + by 2

for the system x = x³ + 2y³ y = x³ (1) prove that V (x,y) = ax²+ by² cannot be used as a Lyapunov function to show stability of the origin for any a and b. [Hint: consider points x = ², y = for small .

(2) what are values of a and b such that V (x,y) = ax⁴ + by⁴ is a Lyapunov function. (3) conclude about the stability of the origin from the Lyapunov function in (2)?

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