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Unformatted text preview: Math 334 Lecture #37 § 9.3: Locally Linear Systems If ~x is a critical point for ~x = f ( ~x ), then the change of variable ~u = ~x- ~x translates the critical point to the origin and the system becomes ~u = f ( ~u ). So we will now assume that ~x = ~ 0 is a critical point for ~x = f ( ~x ). We will further assume that the critical point ~x = ~ 0 is an isolated critical point: there is a circle of positive radius about the origin within which the only critical point is the origin. Now we suppose that that we can write f ( ~x ) = A~x + g ( ~x ) where A is a 2 × 2 matrix with nonzero determinant, and g has continuous partial derivatives. For the nonlinear system ~x = f ( ~x ) to be “close” to the linear system ~x = A~x requires that g ( ~x ) be small for ~x “close” to ~ 0: this is quantified by k g ( ~x ) k k ~x k → 0 as ~x → ~ . When this holds, we say that ~x = f ( ~x ) is a locally linear system at the critical point ~x = ~ 0, and ~x = A~x is the local linear system, or the linearization of ~x = f ( ~x ) at the critical point ~x = ~ 0. Outcome A: Finding the Local Linear System at an Isolated Critical Point . For the sys- tem x = F ( x, y ), y = G ( x, y ) suppose that the first and second order partial derivatives of F and G are continuous near a critical point ( x , y )....
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