hw2cds140b

hw2cds140b - 6). (a) Introduce a new time scale = t and...

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1 CDS 140b: Homework Set 2 Due by Thursday, January 29, 2009. 1. Consider the system ˙ x = - y + x ( r 4 - 3 r 2 + 1) ˙ y = x + y ( r 4 - 3 r 2 + 1) where r 2 = x 2 + y 2 . (a) Show that ˙ r < 0 on the circle r = 1 and ˙ r > 0 on r = 2. Use the Poincar´ e- Bendixson theorem to show that there exists a limit cycle in the annular region A 1 = { ( x,y ) : 1 < r < 2 } . Hint: for this part of the exercise, a slightly modified version of the Poincar´ e- Bendixson theorem has to be used. See remark 1 in the notes. (b) Show that the origin is an unstable fixed point of the system and use this together with the fact that ˙ r < 0 on r = 1 to conclude (again by the Poincar´ e-Bendixson theorem) that there exists a limit cycle in the annular region A 2 = { ( x,y ) : 0 < r < 1 } . (c) Convert the system to polar coordinates, find the explicit expression for both limit cycles and discuss their stability. 2. Show that the Li´ enard system defined by g ( x ) = x and F ( x ) = x 3 - x x 2 + 1 , i.e. ˙ x = y - F ( x ), ˙ y = - g ( x ), has a unique stable limit cycle. 3. Write out the details for the Poincar´ e-Lindstedt method applied to the Duffing oscil- lator as described in the notes (
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Unformatted text preview: 6). (a) Introduce a new time scale = t and rewrite the ODE in this new time scale. (b) Introduce series expansions for x and and derive equations for x and x 1 . What are the initial conditions for x and x 1 ? (c) Solve the equation for x 1 and use the freedom in the series expansion for to eliminate secular terms. Hint: use the trigonometric identity cos 3 = 3 cos + cos 3 4 . (d) Show that the zeroth-order solution for the Dung oscillator is given by x ( t ) = a cos(1 + 3 8 a 2 ) t. 4. Apply the averaging method to the Dung oscillator x + x + x 3 = 0 to nd a rst approximation ( i.e. the x term in the expansion) to the true solution. First, derive the averaged equations. Then, integrate these equations for a general set of initial conditions and compare the solution with the approximation obtained in class using the Poincar e-Lindstedt method....
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.

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