Thus the initial point must be on the convergence line +T5,forr∈R Given the initial point(o, yo)on the convergence line, (3. 18)implies the adjustment path t y(t) which converges to (,). The non-convergence line is defined by +T(,forr∈R y Theorem 3.6. For the linear system A(x-) where a E R2 and A E RX2, suppose the system is saddle-point stable with A<0 Then, it has a negative eigenvalue A<0 and a positive eigenvalue u>0. Let s and c be eigenvectors of A and A, respectively. Then, the convergence line is x=+T5, and the non-convergence line is R If the initial point o is on the convergence line, the convergent path is and if the initial point o is on the non-convergence line, the non-convergent path is Example 3.11. Draw the phase diagram for the following system (x-)+10(y-0), 20 3(y-0)Thus, the initial point must be on the convergence line: ⎛ ⎜⎝ x y ⎞ ⎟⎠ = ⎛ ⎜⎝ x¯ y¯ ⎞ ⎟⎠ + τ ξ, for τ ∈ R. Given the initial point (x0, y0) on the convergence line, (3.18) implies the adjustment path: ⎛ ⎜⎝ x(t) y(t) ⎞ ⎟⎠ = ⎛ ⎜⎝ x¯ y¯ ⎞ ⎟⎠ + ⎛ ⎜⎝ x0 − x¯ y0 − y¯ ⎞ ⎟⎠ eλt , which converges to (¯x, y¯). The non-convergence line is defined by ⎛ ⎜⎝ x y ⎞ ⎟⎠ = ⎛ ⎜⎝ x¯ y¯ ⎞ ⎟⎠ + τ ζ, for τ ∈ R. Theorem 3.6. For the linear system x˙ = A(x − x¯), where x ∈ R2 and A ∈ R2×2, suppose the system is saddle-point stable with |A| < 0. Then, it has a negative eigenvalue λ < 0 and a positive eigenvalue μ > 0. Let ξ and ζ be eigenvectors of λ and μ, respectively. Then, the convergence line is x = ¯x + τ ξ, τ ∈ R, and the non-convergence line is x = ¯x + τ ζ, τ ∈ R. If the initial point x0 is on the convergence line, the convergent path is x = ¯x + (x0 − x¯)eλt ; and if the initial point x0 is on the non-convergence line, the non-convergent path is x = ¯x + (x0 − x¯)eμt .  Example 3.11. Draw the phase diagram for the following system: x˙ = −(x − x¯) + 10(y − y¯), y˙ = 20(x − x¯) + 3(y − y¯).  3—9