Nonstable Path with A>0. If (d1+d1)2<4 Al, the two eigenvalues are i6 d1+d i6 2 where AJ-(td )2. By(3. 18), the adjustment path is +(5ex,(e")(,() yt) 0-9 nce eAt=ez (cos(0t)-isin(et) oHt=e 2t [cos(et)+isin(et) the convergence is totally determined by d1 +di 1. If d1 +d1=0, the adjustment path will be a cycle(a vortex with uniform fluctua- tions 2. If d1 +di<0, the adjustment path is convergent but with cycles(damped fluctua- tions 3. If d1 +di>0, the adjustment path is non-convergent and with cycles(explosive flu uctuations If(d1+d1)224 Al, since [A>0, the signs of A and u are the same as the sign of d1+d1 1. If d1+di>0, an adjustment path starting from any point is non-stable 2. If d1 +d1<0, an adjustment path starting from any point is convergent Stable Path with A=0. If A=0, one of the eigenvalues must be zero and the two demarcation lines defined by A =0 are merged into one y-y If d1+d1>0, a convergent path must start from the convergence line and it stays at the initial point forever If d1+d1<0, any sequence is convergent, but the limit is generally not(,g)Nonstable Path with |A| > 0. If (d1 + d1)2 < 4 |A| , the two eigenvalues are λ = d1 + d1 2 − iθ, μ = d1 + d1 2 + iθ, where θ ≡ t |A| − d1+d1 2 2 . By (3.18), the adjustment path is ⎛ ⎜⎝ x(t) y(t) ⎞ ⎟⎠ = ⎛ ⎜⎝ x¯ y¯ ⎞ ⎟⎠ + ξeλt , ζeμt  (ξ, ζ) −1 ⎛ ⎜⎝ x0 − x¯ y0 − y¯ ⎞ ⎟⎠ . Since eλt = e d1+d1 2 t [cos(θt) − isin(θt)] , eμt = e d1+d1 2 t [cos(θt) + isin(θt)] , the convergence is totally determined by d1 + d1. 1. If d1 + d1 = 0, the adjustment path will be a cycle (a vortex with uniform fluctua￾tions). 2. If d1 + d1 < 0, the adjustment path is convergent but with cycles (damped fluctua￾tions). 3. If d1 + d1 > 0, the adjustment path is non-convergent and with cycles (explosive fluctuations). If (d1 + d1)2 ≥ 4 |A| , since |A| > 0, the signs of λ and μ are the same as the sign of d1 + d1. 1. If d1 + d1 > 0, an adjustment path starting from any point is non-stable. 2. If d1 + d1 < 0, an adjustment path starting from any point is convergent. Stable Path with |A| = 0. If |A| = 0, one of the eigenvalues must be zero and the two demarcation lines defined by A ⎛ ⎜⎝ x − x¯ y − y¯ ⎞ ⎟⎠ = 0 are merged into one. If d1 + d1 > 0, a convergent path must start from the convergence line and it stays at the initial point forever. If d1 + d1 < 0, any sequence is convergent, but the limit is generally not (¯x, y¯). 3 — 10