Section 6.1 
a) Find the critical points for the following autonomous systems. 

b) Use pplane7.m in matlab to see the phase portrait.   

   i) Based on the phase portrait describe the critical points as:   

            spiral point, proper node, improper node, saddle point, center 

   ii) Give their stability:   

            stable, asymptotically stable, unstable 

   iii) And state if they are: 

           sink, source, or neither 

1)     

Answer:  a) (0, 0)    b) saddle point, unstable, neither 

2)  

3)     

Answer:  a) (-1, 1)    b) center, stable, neither 

4)  

5)     

Answers :  a) (-2, 1)  b) spiral point, unstable, source 

                 a) (2, -1)  b) saddle point, unstable, neither 

6)  

7)     

Answer:  a) (0, 0)  b) spiral point, unstable, source 

               a) (-2, -1)  b) saddle point, unstable, neither 

               a) (2, 1)  b) saddle point, unstable, neither 

8)

Section 6.2 

Apply Theorem 1 to determine the type of the critical point (0, 0) of the linear systems and whether it is asymptotically stable, stable, or unstable.  Describe the point as: 

Real, unequal, same sign	Improper node
Real, unequal, opposite sign	Saddle point
Real, equal	Proper or improper node
Complex conjugate	Spiral point
Pure imaginary	Center

1)  

Answer:  Asymptotically stable node 

2)  

3)  

Answer:  Unstable saddle point 

4)  

5)  

Answer:  Asymptotically stable node

Find the single critical point of the almost linear systems and apply Theorem 2 to determine the type and stability (asymptotically stable, stable, or unstable).  Describe the point as: 

	Stable node or spiral point
	Unstable node or spiral point
	Unstable improper node
	Stable spiral point
	Stable or unstable, center or spiral point

11)  

Answer:  Asymptotically stable node:  (2, 1) 

12)  

13)  

Answer:  Unstable saddle point:  (2, 2) 

14)  

15)  

Answer:  Asymptotically stable spiral point:  (1, 1)