Math 260 Classifying fixed points of nonlinear systems in the plane

Save your work regularly--you never know when Mathematica may crash.  

Email your completed file to tleise@amherst.edu by Monday 4pm.

Plotting vector fields in the plane in Mathematica

We can get a good idea of what trajectories look like by plotting the velocity field ((OverDot[x]),(OverDot[y])) for a system of differential equations:

VectorPlot[{Sin[x],Cos[y]},{x,-Pi,Pi},{y,-Pi,Pi}]

Exercises

Exercise 1.  Consider the system (OverDot[x])=f(x,y)=sin(x), (OverDot[y])=g(x,y)=cos(y).
    (a) Based on the vector field given above, how would you classify the fixed point (0,π/2)?
    (b) Find the eigenvalues of the matrix A=(({{((∂f)/(∂x)), ((∂f)/(∂y))}, {((∂g)/(∂x)), ((∂g)/(∂y))}}))  evaluated at (x,y)=(0,π/2).  Classify the fixed point (0,π/2) based on these eigenvalues.
    (c) Based on the vector field given above, how would you classify the fixed point (0,-π/2)?
    (d) Find the eigenvalues of the matrix A=(({{((∂f)/(∂x)), ((∂f)/(∂y))}, {((∂g)/(∂x)), ((∂g)/(∂y))}}))  evaluated at (x,y)=(0,-π/2).  Classify the fixed point (0,-π/2) based on these eigenvalues.

Exercise 2.  Consider the system (OverDot[x])=f(x,y)=x-y, (OverDot[y])=g(x,y)=1-Exp[x].
    (a) Find the fixed point and draw the vector field near this fixed point.
    (b) Find the eigenvalues of the matrix A=(({{((∂f)/(∂x)), ((∂f)/(∂y))}, {((∂g)/(∂x)), ((∂g)/(∂y))}}))  evaluated at the fixed point.  
    (c) Classify the fixed point based on these eigenvalues.

Exercise 3.  Consider the system (OverDot[x])=f(x,y)=x(2-x-y), (OverDot[y])=g(x,y)=x-y. 
    (a) Find the fixed points (you may use the Solve command) and draw the vector field containing both fixed points.
    (b) Find the eigenvalues of the matrix A=(({{((∂f)/(∂x)), ((∂f)/(∂y))}, {((∂g)/(∂x)), ((∂g)/(∂y))}}))  evaluated at each fixed point (so you'll have two different matrices).  
    (c) Classify each fixed point based on the eigenvalues of the corresponding matrix.

Exercise 4.  Consider the system (OverDot[x])=f(x,y)=y+y^2, (OverDot[y])=g(x,y)=-x/2+y/5-x y+6/5y^2.
     (a) Find the fixed points (you may use the Solve command) and draw the vector field containing both fixed points.
    (b) Find the eigenvalues of the matrix A=(({{((∂f)/(∂x)), ((∂f)/(∂y))}, {((∂g)/(∂x)), ((∂g)/(∂y))}}))  evaluated at each fixed point (again you'll have two different matrices).  
    (c) Classify each fixed point based on the eigenvalues of the corresponding matrix.