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445/545
Test #2
TEST #2 Study Guide
11/03/01
1
Problem #1:
Linearizing a non-linear first order system at its equilibrium points.
(a)-[3 pts.]-Locate the equilibrium points of a given non-linear system: x  f ( x , y ), y  g( x , y ) .
(b)-[3 pts.]-Find the matrix of the linearized system at each equilibrium point.
(c)-[6 pts.]-Find the eigenvalues of the matrix of the linearized system at each equilibrium point.
(d)-[3 pts.]-Use the results of part (c) to classify each of the equilibrium points.
(e)-[3 pts.]- Sketch the direction field of the given system along the positive y-axis, and along the negative yaxis.
(f)-[3 pts.]- Sketch the direction field of the given system along the positive x-axis, and along the negative xaxis.
(g)-[4 pts.]- Sketch the direction field of the given system along each solution plotted on the graph below.
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Problem #2:
Classifying the equilibrium points of a non-linear first order system .
The phase portrait of the non-linear system x  f ( x , y ), y  g( x , y ) is sketched.
(a)-[3 pts.]- Where are the equilibrium points of the given system located ?
(b)-[4 pts.]- Classify each equilibrium point of the given system.
(c)-[6 pts.]- Discuss the eigenvalues of the linearized system at each equilibrium point.
Are they real or complex ? What are their algebraic signs if they are real ?
What can you say about the real parts of any complex eigenvalue ?
Given a sketch of the phase portrait of the non-linear system x  f ( x , y ), y  g( x , y ) :
(d)-[3 pts.]-Locate the equilibrium points of the given system located ?
(e)-[3 pts.]-Classify each equilibrium point of the given non-linear system.
(f)-[6 Ps.]-Discuss the eigenvalues of the linearized system at each equilibrium point.
Are they real or complex ? What are their algebraic signs if they are real ?
What can you say about the real parts of any complex eigenvalue ?
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Problem #3:
Stable and Unstable Manifolds
(a)-[8 pts.]- Find the solution of the following initial value problem:
x'  f ( x , y ), y'  g( x , y ), x( 0 )  x 0 , y( 0 )  y 0
2
where f ( x, y)  x  y , and g ( x, y )   y
(b)-[6 pts.]-Use the result of part (a) to find an equation for the global stable manifold of the given system at the
equilibrium point (0,0).
(c)-[4 pts.]- Determine the unstable manifold for the given system at (0,0).
(d)-[4 pts.]- Use the results of (a) and (b) to prove that that the stable manifold of the given system is invariant
under the flow of the vector field ( f ( x , y ), g( x , y )) : i.e. prove that any solution starting at a point on the stable
manifold at t  0 , remains on the stable manifold for all t .
(d)-[3 pts.]- Sketch the stable and unstable manifolds on the figure below.
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Problem #4:
Hartman-Grobman Theorem
(a)-[5 pts.]- Find the inverse of the given coordinate transformation
y1  F ( x1 , x2 , x3 ), y 2  G( x1 , x2 , x3 ), y3  H ( x1 , x2 , x3 ) .
(b)-[5 pts.]- Use the result of part (a) to reduce the given non-linear system
x1  f ( x1 , x2 , x3 ), x 2  g( x1 , x2 , x3 ) , x 3  h( x1 , x2 , x3 )
to the linear system y  Ay where
 f x1 ( 0 ,0 ,0 ) f x2 ( 0 ,0 ,0 ) f x3 ( 0 ,0 ,0 )
 y1 




y   y 2  , and A   g x1 ( 0 ,0 ,0 ) g x2 ( 0 ,0 ,0 ) g x3 ( 0 ,0 ,0 ) .
 hx ( 0 ,0 ,0 ) hx ( 0 ,0 ,0 ) hx ( 0 ,0 ,0 ) 
 y 3 
2
3
 1

(c)-[5 pts.]- Using the result of part (b) what conclusion can be drawn about the equilibrium
point (0,0,0) of the given non-linear system? Briefly explain your answer.
(d)-[10 pts.]- Use the result of (a) to find equations for the stable and unstable manifolds of
the given system of o.d.e.'s at (0,0,0).
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Problem #5: Using Liapunov Functions to Classify Equilibrium Points
(a)-[10 pts.]- Find a Liapunov function for the sytem x  f ( x , y ), y  g( x , y ) at (0,0).
Hint: Try V ( x, y )  Polynomial in x and y with appropriately chosen coefficients.
(b)-[10 pts.]- Using the result of part (a) what conclusion can be drawn about the equilibrium point of the given
system at (0,0)?
(c)-[5 pts.]- Calculate the eigenvalues of the linearized system at (0,0) and briefly explain why no conclusion
about the equilibrium point of the non-linear system can be drawn from the behavior of the solutions of the
linearized system near (0,0).
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Problem #6:
Hamiltonian Systems
(a)-[4 pts.]-Verify that a given system x  f ( x , y ), y  g( x , y ) is Hamiltonian.
(b)-[8 pts.]-Find the Hamiltonian of the given system.
(c)-[6 pts.]-Locate and classify the equilibrium points of the given system.
(d)-[7 pts.]-Which of the phase portraits sketched below is the phase portrait of the given
system Figure1 or Figure 2 ?
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445/545
Test #2
11/03/01
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