Economic Model With Investment

The dynamical system

represents an economic model of a national economy in which the investment is given by and is a parameter.

1. Show that the system has a single equilibrium point at the origin.

3. Show that the equilibrium point is a stable focus for and an unstable focus for .

5. Consider a Lyapunov function of the form

6. By choosing a and b appropriately show that for the case , can be written in the form

   Hence show that except on the line . (Hint: this can be done easily by plotting a graph. Try proving it analytically).

7. What can you deduce about the stability of the origin for the case from the results of (c)?

9. Use Maple to verify that for the origin is a stable focus. Show also that for

the origin is an unstable focus surrounded by a stable limit cycle whose size increases as r is increased.