lecture2B

lecture2B - Classification of Linear Systems I Given a...

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Unformatted text preview: Classification of Linear Systems I Given a linear system ˙ x = A x , the desired straight-line solutions ( x = e λt v ) exist if v is an eigenvector of A with eigenvalue λ , A v = λ v . I Recall: eigenvalues of A is given by characteristic equation det( A- λI ) which has solutions λ 1 = τ + p τ 2- 4 4 2 , λ 2 = τ- p τ 2- 4 4 2 where τ = trace( A ) = a + d and 4 = det( A ) = ad- bc . I If λ 1 6 = λ 2 (typical situation), eigenvectors its v 1 and v 2 are linear independent and span the entire plan. General Solution for Initial Value Problem I Can write any initial condition as a linear combination x = c 1 v 1 + c 2 v 2 I General solution is given by x ( t ) = c 1 e λ 1 t v 1 + c 2 e λ 2 v 2 I Solve IVP ˙ x = x + y, ˙ y = 4 x- 2 y with ( x , y ) = (2 ,- 3) and draw phase portrait. Both Eigenvalues are Real and Different I Have opposite signs: Fixed point is a saddle . I Same sign ( negative , positive ): nodes ( stable , unstable ). Eigenvalues are Complex Conjugates I Eigenvalues...
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This note was uploaded on 01/04/2012 for the course CDS 280 taught by Professor Marsden,j during the Fall '08 term at Caltech.

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lecture2B - Classification of Linear Systems I Given a...

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