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Phase - A Catalogue of Phase Portraits of 2-D Linear Flows...

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A Catalogue of Phase Portraits of 2-D Linear Flows Consider all (real variable) solutions of 2-D system bracketleftbigg x ( t ) y ( t ) bracketrightbigg = A bracketleftbigg x ( t ) y ( t ) bracketrightbigg , where A is a constant 2 × 2 real matrix. By sketching the phase portrait of this system, we mean to draw several representative solution curves on the x - y plane to display typical asymptotic behavior, especially the solution behavior as t → ∞ and t → −∞ . Equilibria and periodic solutions should be displayed. The stability of equilibria and periodic solutions should be easily read off from the picture. The structure of eigenvalues and (generalized) eigenvectors of matrix A gives the solution formula, completely determines dynamic behavior of the system, and therefore will also guide our classification of phase portraits.
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The eigenvalues of A are: λ 1 < λ 2 < 0. Solution Method: Prepare an eigenvector u 1 for λ 1 : that is, ( A λ 1 I ) u 1 = 0 , u 1 negationslash = 0. Prepare an eigenvector u 2 for λ 2 : that is, ( A λ 2 I ) u 2 = 0 , u 2 negationslash = 0. The general solutions of the differential system are bracketleftbigg x ( t ) y ( t ) bracketrightbigg = C 1 e λ 1 t u 1 + C 2 e λ 2 t u 2 , where C 1 and C 2 are free parameters. Example. A = 1 19 bracketleftbigg 58 5 4 37 bracketrightbigg , λ 1 = 3 , u 1 = bracketleftbigg 5 1 bracketrightbigg , λ 2 = 2 , u 2 = bracketleftbigg 1 4 bracketrightbigg .
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The eigenvalues of A are: λ 1 > λ 2 > 0. Solution Method: Prepare an eigenvector u 1 for λ 1 : that is, ( A λ 1 I ) u 1 = 0 , u 1 negationslash = 0. Prepare an eigenvector u 2 for λ 2 : that is, ( A λ 2 I ) u 2 = 0 , u 2 negationslash = 0. The general solutions of the differential system are bracketleftbigg x ( t ) y ( t ) bracketrightbigg = C 1 e λ 1 t u 1 + C 2 e λ 2 t u 2 , where C 1 and C 2 are free parameters. Example. A = 1 19 bracketleftbigg 58 5 4 37 bracketrightbigg , λ 1 = 3 , u 1 = bracketleftbigg 5 1 bracketrightbigg , λ 2 = 2 , u 2 = bracketleftbigg 1 4 bracketrightbigg .
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The eigenvalues of A are: λ 1 < 0 < λ 2 . Solution Method: Prepare an eigenvector u 1 for λ 1 : that is, ( A λ 1 I ) u 1 = 0 , u 1 negationslash = 0.
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