Given the homogenous system of linear differentialequations
Dx¯= Ax¯
Where A=[ai,j] is a real valued, 2Ã2 coefficient matrix, we canclassify the equilibrium solution x¯= 0¯ in one of the severaldifferent ways:saddle, node, spiral, and center, in addition tostable, asymptotically stable, or unstable. Refferring to thesegenerically as properties of the equilibrium solution, we saythat:
Definition: a property of the equilibrium solution persistsunderperturbation if there exists a real number ? > 0 such that, forA? = [ai,j + ?i, j], the equilibrium solution of
Dx¯= A?x¯
Has the same property as the equilibrium solution of Dx¯=Ax¯ forall ?i,j ? ( -?,?).
With properties of the equilibrium solution persist underperturbation? Prove your claim
Note that x¯ means a vector because could get an arrow pointinglooking symbol
Also ai,j shows that I and j are lower cased on bottom ..if u knowwhat I mean.. thanx for the help