M334Lec37 - Math 334 Lecture#37 9.3 Locally Linear Systems...

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Math 334 Lecture #37 § 9.3: Locally Linear Systems If x 0 is a critical point for x = f ( x ), then the change of variable u = x - x 0 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 ) = Ax + 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 = Ax requires that g ( x ) be small for x “close” to 0: this is quantified by g ( x ) x 0 as x 0 . When this holds, we say that x = f ( x ) is a locally linear system at the critical point x = 0, and x = Ax 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 0 , y 0 ).
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