A Theorem in the F & M Deiner Style

Theorem 1.5.1: Microscopic Stability

Let f [x, y] and g[x, y] be smooth functions with f[x_e, y_e] = g[x_e, y_e] = 0. The flow of

dx/dt = f[x, y]
dy/dt = g[x, y]

under infinite magnification at (x_e, y_e) appears the same as the flow of its linearization

( du ) = ( a a ) ( u ) -- ... b b dv x y -- dt,

a_x = ∂f/∂x[x_e, y_e], a_y = ∂f/∂y[x_e, y_e], b_x = ∂g/∂x[x_e, y_e], b_y = ∂g/∂y[x_e, y_e]

Specifically, if our magnification is 1/δ,for δ≈0, and our solution starts in our view, Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view, Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view, Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view, Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view,

(x[0] −x_e, y[0] −y_e) = δ · (a, b)

for limited a and b and if (u[t], v[t]) satisfies the linear equation and starts at (u[0], v[0]) = (a, b),then Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view, Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view, Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view, Specifically, if our magnification is 1/δ,forδ0, and our solution starts in our view,

(x[t] −x_e, y[t] −y_e) = δ · (u[t], v[t]) + δ · (ε_x[t], ε_y[t])

where (ε_x[t], ε_y[t]) ≈ (0, 0) for all limited t.

This is a fun and easy theorem - the pictures work even when the eigenvalues of the linear system are zero! where nothing moves in the linear view.  For more details see p.136.

Created by Mathematica  (July 2, 2004)