1550078481-Ordinary_Differential_Equations__Roberts_

Answers to Selected Exercises

5.

r s E Possible Arms Race(s)

(^0 0) + md
0 0 rar

• md
  md, sar, rar


• 
  
  
  
  • 
    
    
    
    • sar
    
    
    
    
  
  
  
  


• 
  
  
  
  • rar
  
  
  
  


• 
  
  
  
  • md, sar
  
  
  
  


• md, sar, rar

Exercises 10 .2 Phase-Plane Portraits

l. >-1 = 1, >-2 = -1, unstable saddle point

3. >-1 = -1, >-2 = -1, asymptotically stable node

5. ,;\ = -1 ± 2i , asymptotically stable spiral point

7. >-1 = -2, >-2 = -4, asymptotically stable node

9. Asymptotically stable node at (0, O); unstable saddle point at (1, 1)


10. Unstable saddle point at (0, O); neutrally stable center at (~, 1)

565

13. Asymptotically stable spiral point at (1, 1); unstable saddle point at

(1, -1)

15. Unstable saddle point at (0, O); asymptotically stable node at (0, - 1) ;
    unstable spiral point at (2, 1)

Exercises 10 .3 Modifie d Rich a rdson's Arms Race Models

l. a. (5, 4)

b. The associated linear system

x' = - lO(x - 5) + 5(y - 4)

y' = 4(x - 5) - 8(y - 4)

has an asymptotically stable node at (5, 4). Therefore, (5, 4) is an

asymptotically stable critical point of the nonlinear system (9).