a. A special case of the Linard equation of is d2u dt2 + du dt + g(u) = 0, where g

Chapter 9, Problem 8

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a. A special case of the Linard equation of is d2u dt2 + du dt + g(u) = 0, where g satisfies the conditions of 5. Letting x = u, y =du/dt, show that the origin is a critical point of the resulting system. This equation can be interpreted as describing the motion of a spring--mass system with damping proportional to the velocity and a nonlinear restoring force. Using the Liapunov function of 5, show that the origin is a stable critical point, but note that even with damping, we cannot conclude asymptotic stability using this Liapunov function. b. Asymptotic stability of the critical point (0, 0) can be shown by constructing a better Liapunov function, as was done in part d of 6. However, the analysis for a general function g is somewhat sophisticated, and we mention only that an appropriate form for V is V( x, y) = 1 2 y2 + Ayg( x) + _ x 0 g(s)ds, where A is a positive constant to be chosen so that V is positive definite and V is negative definite. For the pendulum problem g( x) = sin x, use V as given by the preceding equation with A = 1 2 to show that the origin is asymptotically stable. Hint: Use sin x = x x3/3! and cos x = 1 x2/2!, where and depend on x, and 0 < < 1 and 0 < < 1 for /2 < x < /2; let x = r cos , y = r sin , and show that V (r cos , r sin ) = 1 2 r2_1 + 1 2 sin 2 + h(r, )_, where |h(r, )| < 1 2 if r is sufficiently small. To show that V is positive definite, use cos x = 1 x2/2 + x4/4!, where depends on x, and 0 < < 1 for /2 < x < /2. In 9 and 10, we will prove part of Theorem 9.3.2: If the critical point (0, 0) of the locally linear system dx dt = a11x+a12 y+F1( x, y), dy dt = a21x+a22 y+G1( x, y) (26) is an asymptotically stable critical point of the corresponding linear system dx dt = a11x + a12 y, dy dt = a21x + a22 y, (27) then it is an asymptotically stable critical point of the locally linear system (26). deals with the corresponding result for instability.