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Unformatted text preview: Professor Tse Page 1 of 7 Spring 2007 Dynamic Systems Midterm Solution Problem 1. Short Answer Questions (a) Model Matching The fate of the Universe “Grabber-Holder Model” Reason. The grabber is “expansion push of Big Bang,” and the holder is “the gravitational attraction force.” The Clubbing Scene “Two Sided Market” Reason. The number of men and women in the club depends on each other. If one of them increases so does the other and vice versa. Note: Partial credit has been given even if you selected another model and made an appropriate explanation for it. (b) True/False (i) False. First, a nonlinear system may not always have an equilibrium point. Second, even if it has an equilibrium point, it may not be determined by analyzing the stability of a linearized system. Think of the case when all = r λ for a continuous system and when 1 | | = λ for a discrete system. (ii) 2 4 ) ( ) ( 2 2 , 1 bc d a d a − − ± + = λ- True. bc a − ± = 2 , 1 λ ; since , we have a spiral out behavior. > a- False. bc − ± = 2 , 1 λ ; since = r λ , we have a cycling behavior. - True. bc a − ± = 2 , 1 λ ; since , we have a spiral in behavior. < a- False. bc a − ± = 2 , 1 λ ; since , we have real eigenvalues. > − bc Professor Tse Page 2 of 7 Spring 2007 Dynamic Systems Midterm Solution Problem 2. The Modern Love Stories (a) Show that this system can be modeled as a continuous linear problem. , where J R R J β α = − = . . β α , are positive. (b) Draw the phase diagram of the system. Unless they both start off seeing each other as friend, they will have a never ending cycle of love and hate. The governing system has a center at . ) , ( (c) Eigenvalues and eigenvectors of the system The corresponding matrix for the problem is: ....
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This note was uploaded on 06/16/2010 for the course MS&E 201 taught by Professor Edisontse during the Spring '08 term at Stanford.
- Spring '08