10Ma2aAnHw7 - , 0) of the following system is...

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Ma 2a Fall 2010 (analytical track) PROBLEM SET 7 Due on Monday, November 22 I. Conservative systems Problem 34.1(iii) (sketch the phase portrait by hand). (X1) Find (approximately) the periods of small oscillations in Problem 34.1(iii) II. Dissipative systems Problems 34.3(ii), 35.2 III. Lyapunov functions and stability of equilibrium solutions (X2) Determine whether the critical point of the following system is stable or un- stable: ˙ x = - x 3 + 2 y 3 , ˙ y = - 2 xy 2 . (X3) Show that the critical point (0
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Unformatted text preview: , 0) of the following system is asymptotically stable: x = 2 y 3-x 5 , y =-x-y 3 + y 5 . [Hint: construct a suitable Lyapunov function of the form ax 2 + y 4 ] IV. Periodic orbits Problem 36.1 [Also, give an argument based on the dissipation of energy] (X4) Draw the phase portrait for the (Rayleigh) equation x + ( x 3- x ) + x = 0 with = 1 and = . 1. Show the limit cycles. 1...
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This note was uploaded on 11/19/2010 for the course ANTH 122 taught by Professor 323 during the Spring '10 term at Centennial College.

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