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Homework07 - Math 404 Homework 6 Example Student Solutions...

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Math 404 - Homework 6 Example Student Solutions February 27, 2011 S: 6.1.13 Draw a phase portrait that has exactly three closed orbits and one fixed point. Answer Figure 1: Phase portrait with exactly 3 closed orbits and 1 fixed point. 1
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S: 6.1.14 (Series approximation for the stable manifold of a saddle point) Recall the system ˙ x = x + exp( - y ), ˙ y = - y from Example 6.1.1. We showed that this system has one fixed point, a saddle at (-1,0). Its unstable manifold is the x - axis, but is unstable manifold is a curve that is harder to find. The goal of this exercise is to approximate this unknown curve. a) Let ( x, y ) be a point on the stable manifold, and assume that ( x, y ) is close to ( - 1 , 0). Introduce a new variable u = x + 1, and write the stable manifold as y = a 1 u + a 2 u 2 + O ( u 3 ). To determine the coefficients, derive two expressions for d y/ d u and equate them. b) Check that your analytical result produces a cure with the same shape as the stable manifold shown in Figure 6.1.4. Answer a) Given this system of equations: dx dt = x + e - y dy dt = - y We know the equilibrum point for this equation is ( - 1 , 0) , so we can introduce a new variable into this equation u = x - 1 Given this we can rewrite our system as: du dt = u - 1 + e - y And dy dt = - y So we would have : dy du = - y u - 1+ e - y But we have another expression for dy du given that we are on the stable manifold. We know that the expression for the stable manifold in the phase plane one will end up being a polynomial of the form y = a 1 u + a 2 u 2 + O ( u 3 ) which is the Taylor polynomial around u = 0, and since this polynomial must pass though (0 , 0) in the phase plane of ( u, y ) so dy du = a 1 + 2 a 2 u + O ( u 2 ). We can equate the two expressions: - y u - 1+ e - y = a 1 + 2 a 2 u What we can do now, is rewrite e - y as 1 - y + y 2 2 + higher order terms. Now we have an approximate expression (which is accurate when we are around the fixed point (0,0) in the (u,y) plane, - y u - 1+1 - y + y 2 2 = a 1 + 2 a 2 u - y u - y + y 2 2 = a 1 + 2 a 2 u 2
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One can plug in the expression we had for y on the manifold: - ( a 1 u + a 2 u 2 + O ( u 3 )) u - ( a 1 u + a 2 u 2 + O ( u 3 ))+ ( a 1 u + a 2 u 2 + O ( u 3 )) 2 2 = a 1 + 2 a 2 u We can expand the denominator, eliminating all terms higher than 2nd order in u (to match terms): - ( a 1 u + a 2 u 2 ) u - ( a 1 u + a 2 u 2 )+ ( a 1 u ) 2 2 = a 1 + 2 a 2 u And - ( a 1 u + a 2 u 2 ) = ( a 1 + 2 a 2 u )( u - a 1 u - a 2 u 2 + ( a 1 u ) 2 2 ) Now expanding more: - a 1 u - a 2 u 2 = a 1 u - a 2 1 u - a 1 a 2 u 2 + a 3 1 2 u 2 + 2 a 2 u 2 - 2 a 2 a 1 u 2 So - a 1 = a 1 - ( a 1 ) 2 a 1 = 0 , 2 0 is the unstable eigendirection given in text, so for the stable manifold, we pick a 1 = 2 now - a 2 = - a 1 a 2 + a 3 1 2 + 2 a 2 - 2 a 1 a 2 Plugging in a 1 : - a 2 = - 2 a 2 + 4 + 2 a 2 - 4 a 2 a 2 = 4 3 So we have equation of the stable manifold for the system to be approxi- mately: y = 2 u + 4 3 u 2 + O ( u 3 ) b) The stable manifold on the phase portrait and the graph of the parabola which approximates the stable manifold resulting from part a are drawn below: The right branch of the parabola resembles the stable manifold of the phase portrait obtained from PPane, as well as the illustration in the textbook.
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