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Unformatted text preview: Classification of Linear Systems I Given a linear system ˙ x = A x , the desired straightline 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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 Fall '08
 Marsden,J
 Linear Algebra, Vector Space, Eigenvalue, eigenvector and eigenspace, Fundamental physics concepts, Eigenfunction, complex conjugates

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