## Homework 3, due Wednesday March 2.

Due Wednesday, March 2, 2005.
goodwine
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### Homework 3, due Wednesday March 2.

1. Consider the motion of a particle of mass m attached to a spring with a nonlinear stiffness
where
and x is the displacement. The differential equation governing the system is
1. Using the total energy as a Lyapunov function, what can you conclude about the stability of the system?
2. If damping is added to the system in the form
what can you conclude about the stability of the system?
2. Using the Lyapunov function
obtain conditions on the coefficients
sufficient for global asymptotic stability of the origin of the Lorenz equations
where
3. Investigate the stability of the origin of
using a Lyapunov function of the form
4. Consider the linear first order system
1. Show that
is the solution of the differential equation.
2. Is the origin a stable fixed point?
3. Is the origin an asymptotically stable fixed point?
4. Is the origin a uniformly asymptotically stable fixed point?
5. Investigate the stability of the origin of
using a Lyapunov function of the form
6. Show that the zero solution of
is stable if
in a neighborhood of the origin.
7. Investigate the stability of the origin of
8. Using the simple example of
work through the proof of case 3 in the table of Lyapunov theorems in Sastry's text.