1Applied Mathematics 105bFebruary 2008 reprint/revision of February 1997 textJ. R. RiceNotes on solutions of (linearized) ode systems near critical pointsLet{x} denote a column ofnfunctionsx1(t),x2(t), ...,xn(t)such that{x}= [x1(t),x2(t),...,xn(t)]T(theTmeans 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 anautonomoussystem of equations (because the functions {f} have no explicitdependence on t). That is equivalent to writingdx1/dt=f1(x1,x2,...,xn) ,dx2/dt=f2(x1,x2,...,xn) ,... ,dxn/dt=fn(x1,x2, ..,xn) .We may think of{x}as a point in a n-dimensional space.(Whenn= 2 , we generally write{x}= [x(t),y(t)]Tand{f} = [f(x,y),g(x,y)]T, so thatdx/dt=f(x,y)anddy/dt=g(x,y) .)Note that the functions{f({x})}uniquely determine the direction of a local trajectorythrough point{x}in the n-dimensional space unless all of the{f}happen to vanishsimultaneously.Forn= 2that corresponds to a trajectory direction through(x,y)in the phaseplane being determined bydy/dx=g(x,y)/f(x,y) , unless bothfandgvanish.Acritical point(orsingular point) is such a point{xcp}for which{f({xcp})}= {0}.Bydoing a Taylor series expansion of the functions{f({x})}about the critical point, neglectingterms beyond the linear inx–xcpin the expansion, and then shifting the origin of the space tothe critical point, i.e., renameingx–xcpasx, we therefore obtain the linearized equations{˙x}=[A] {x}
