MATH503 hw5 - , 0) is (i) a stable spiral if <...

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Homeworks (Math 503) T1 : Div, grad, curl and all that , by Schey (4th edition) T2 : Calculus of variations T3 : Nonlinear dynamics and chaos , by Strogatz Note: Detail your work to receive full credit Homework#5: (due Wed Nov 17) 1. Solve, sketch the phase portrait and classify the fixed point of the following systems: (a) ˙ x = 5 x + 3 y , ˙ y = - 4 x - 3 y . (b) ˙ x = x - 2 y , ˙ y = 3 x - 4 y . (c) ˙ x = x - 5 y , ˙ y = x - 3 y . 2. Consider the system ˙ x = - x + ±y , ˙ y = x - y . Show that the fixed point (0 , 0) is (i) a stable spiral if ± < 0 and (ii) a stable node if 0 ± < 1. Compared to the unperturbed system with ± = 0, do small perturbations ± 6 = 0 change the type and stability of the fixed point (0 , 0)? 3. Consider the system ˙ x = ±x - y , ˙ y = x + ±y . Show that the fixed point (0
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Unformatted text preview: , 0) is (i) a stable spiral if &lt; 0, (ii) a center if = 0 and (iii) an unstable spiral if &gt; 0. Compared to the unperturbed system with = 0, do small perturbations 6 = 0 change the type and stability of the xed point (0 , 0)? 4. The motion of a damped harmonic oscillator is described by m x + b x + kx = 0 where b &gt; 0 is the damping constant. (a) Rewrite the equation as a two-dimensional linear system. (b) Classify the xed point at the origin. Consider all the dierent cases depending on the sign of the discriminant of the eigenvalues. (c) Relate your results to the standard notions of overdamped and underdamped vibrations....
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This note was uploaded on 02/20/2011 for the course MATH 503 taught by Professor Schleiniger,g during the Fall '08 term at University of Delaware.

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