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Unformatted text preview: Fixed Points and Linearization I Suppose ( x * , y * ) is fixed point, linearized system is ˙ u ˙ v = ∂f ∂x ∂f ∂y ∂g ∂x ∂g ∂y ( x * ,y * ) u v where u = x x * , v = y y * . Jacobian matrix . I If fixed point for linearized system is not one of borderline cases, linearized system give a qualitatively correct picture near ( x * , y * ). I Borderline cases can be altered by small nonlinear terms. Example 6.3.1 I Consider ˙ x = x + x 3 , ˙ y = 2 y. I 3 fixed points: (0 , 0) (stable node ), (1 , 0) , ( 1 , 0) ( saddles ) by analyzing linearized system . I Not borderline cases. Fixed points for nonlinear system is similar to linearized system . Example 6.3.2 I Consider ˙ x = y + ax ( x 2 + y 2 ) , ˙ y = x + ay ( x 2 + y 2 ) . I Linearized system predicts: (0 , 0) is a center for all a . I In polar coordinates ˙ r = ar 3 , ˙ θ = 1 . I In fact, (0 , 0) is a spiral (stable if a < 0, unstable if a > 0)....
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 Spring '10
 Cline
 Fundamental physics concepts, Stability theory, Dynamical systems, borderline cases

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