Nonlinear Damping In the analysis of free, damped motion in Section 5.1 we assumed that

Chapter 10, Problem 17

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Nonlinear Damping In the analysis of free, damped motion in Section 5.1 we assumed that the damping force was proportional to the velocity x. Frequently, the magnitude of this damping force is proportional to the square of the velocity, and the new differential equation becomes . (a) Write the second-order differential equation as a plane autonomous system, and find all critical points. (b) The system is called overdamped when (0, 0) is a stable node and is called underdamped when (0, 0) is a stable spiral point. Physical considerations suggest that (0, 0) must be an asymptotically stable critical point. Show that the system is necessarily underdamped