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.

  2.  

     

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

  4.  

     

  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)?

  8.  

     

  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.