Homework3_2010

Homework3_2010 - ME580 Nonlinear Systems Home work #3 Due:...

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ME580 Nonlinear Systems Home work #3 Due: October 7, 2010 1. Three different systems have an equation of motion of the form dx(t)/dt = a(t)x(t), where a(t) is of period T > 0. Find the period propagator K, and determine the stability of the equilibrium point for all real values of the constants b and c, when a(t) is given by: i) a(t) = 0 (0 < t < T/2), = b (T/2 < t < T); ii) a(t) = 3b+ct (0 < t < T); iii) a(t) = b t-T/2 ( 0 < t < T). 2. For the following systems, a. 3 30 x x x , b. 3 x x x , c. 2 x x x , i) Find the equilibrium points; ii) Find the local phase portraits around the equilibrium points, and characterize the nature of the equilibrium points in terms of a center or a saddle; iii) Find the potential functions for the systems in (a), (b), and (c), and use them to draw the complete phase portraits; and iv) For the regions in the phase plane where periodic motions are possible for these systems, use integration of the equations describing the level curves to find the relation between the period of oscillation and the corresponding amplitude of motion.
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Homework3_2010 - ME580 Nonlinear Systems Home work #3 Due:...

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