X = rX has a stable equilibrium point at X = 0 if and only if r

why does x' has a stable equilibrium ONLY if r

Photos - 2022-01-25.png 65% Fullscreen X File | C:/Users/JQ/Downloads/Modeling%20Life%20The%20Mathematics%20of... & Co G . .. 1.00 Equilibrium Points First of all, let's discuss equilibrium points. If we think about one-dimensional linear vector fields, then we are talking about either X' = rX or X' =-rx (assuming r > 0) It is clear that the only equilibrium points these systems can have are X = 0. But what about two-dimensional or even n-dimensional cases? In the n-dimensional case, if we are looking for equilibrium points, we are looking for solutions to X' = 0 = f(X) which implies X; = 0 = fi (X1, X2. . . . . Xn) X' =0 = f2(X1, X2. . .. . Xn) Xh = 0 = f. (X1. X2. . . . . Xn) where fi. f2. . ... fo are all linear functions R" - R. How many solutions can this set of equations have? Here, a theorem from elementary algebra comes to the rescue: setting n linear functions of n unknowns equal to zero can have only one solution, and that is X1 = X2 = . . . = Xn = 0 (We can find this by using the first equation to eliminate X, in terms of the other variables, then using the second equation to eliminate X2, and finally we get an equation of the form aX, = 0. which can have only the solution X1 = X2 = . . . = Xn = 0.) A linear system of differential equations has a unique equilibrium point, at X1 = X2 = . ..= Xn = 0 Stability Having found the equilibrium point, we now need to determine its stability. In the one-dimensional case, we have already seen that X" = rX has a stable equilibrium point at X = 0 if and only if r