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12/08/11 - Phase-Plane Portraits and the Trace-Determinant Line
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 | The solutions of systems of two dependent linear first-order homogeneous systems with constant coefficients are determined by the eigenvalues and eigenvectors of the matrix. These, in turn, are functions of the trace and determinant of the matrix. Hence it may not be surprising that we can classify the different types of solutions based on the values of the trace and determinant.
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| Note that there are four basic types of solutions (based on the Trace T and the Determinant D):
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| 1 -- Distinct real eigenvalues
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where v1 and v2 are the eigenvectors corresponding to λ1 and λ2, respectively.
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where
and
where w is the complex eigenvector corresponding to λ.
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| 3 -- Two identical real eigenvalues and two distinct eigenvectors
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where λ = T/2 is the eigenvalue and v is the vector corresponding to the initial condition:
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| 4 -- Two identical real eigenvalues but only one eigenvector
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| The Trace-Determinant (TD) Plane
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| D = T2/4 (two identical eigenvalues)
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| Solve the problems on the worksheet below:
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