GT2 Eigenvalue Classification and Laplace

Laplace of tⁿ AKA (L(tⁿ)

n!/sⁿ⁺¹

Laplace for delta/dirac

Integral ( 0 —> inf) of d(t-t_o) e^-st dt = e^-st_o

Both eigenvalues real and positive

unstable node

Both eigenvalues real and negative

(asymptotically) stable node

Both eigenvalues real, but opposite signs

unstable saddle

both eigenvalues are purely imaginary (-bi and bi)

stable center

both eigenvalues are complex imaginary (a ± bi) with negative a

asymptotically stable spiral

both eigenvalues are complex imaginary (a ± bi) with positive a

unstable spiral

Notation for approximating linear system for critical points

(x’ ; y’) = (F_x(x₁) F_y(y₁); G_x(x₁) G_y(y₁))(x; y)