02d_critical_pts_ode - Applied Mathematics 105b February...

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1 Applied Mathematics 105b February 2008 reprint/revision of February 1997 text J. R. Rice Notes on solutions of (linearized) ode systems near critical points Let { x } denote a column of n functions x 1 ( t ), x 2 ( t ), ..., x n ( t ) such that { x } = [ x 1 ( t ), x 2 ( t ), ..., x n ( t )] T (the T means transpose, i.e., interchange rows and columns, and here shifts the row to a column). We suppose that the { x } satisfy the system of ode's of the form { ˙ x } = { f ({ x }) } , called an autonomous system of equations (because the functions { f } have no explicit dependence on t). That is equivalent to writing dx 1 / dt = f 1 ( x 1 , x 2 ,..., x n ) , dx 2 / dt = f 2 ( x 1 , x 2 ,..., x n ) , ... , dx n / dt = f n ( x 1 , x 2 , .., x n ) . We may think of { x } as a point in a n-dimensional space. (When n = 2 , we generally write { x } = [ x ( t ), y ( t )] T and { f } = [ f ( x,y ), g ( x,y )] T , so that dx/dt = f ( x,y ) and dy/dt = g ( x,y ) .) Note that the functions { f ({ x }) } uniquely determine the direction of a local trajectory through point { x } in the n-dimensional space unless all of the { f } happen to vanish simultaneously. For n = 2 that corresponds to a trajectory direction through ( x,y ) in the phase plane being determined by dy/dx = g ( x,y )/ f ( x,y ) , unless both f and g vanish. A critical point (or singular point ) is such a point { x cp } for which { f ({ x cp }) } = { 0 }. By doing a Taylor series expansion of the functions { f ({ x }) } about the critical point, neglecting terms beyond the linear in x x cp in the expansion, and then shifting the origin of the space to the critical point, i.e., renameing x x cp as x , we therefore obtain the linearized equations { ˙ x } = [ A ] { x }
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