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MATH503 hw5

# MATH503 hw5 - 0 is(i a stable spiral if ±< 0(ii a center...

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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 ± < 0, (ii) a center if ± = 0 and (iii) an unstable spiral if ± > 0. Compared to the unperturbed system with ± = 0, do small perturbations ± 6 = 0 change the type and stability of the ﬁxed point (0 , 0)? 4. The motion of a damped harmonic oscillator is described by m ¨ x + b ˙ x + kx = 0 where b > 0 is the damping constant. (a) Rewrite the equation as a two-dimensional linear system. (b) Classify the ﬁxed point at the origin. Consider all the diﬀerent 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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