c$_{1}$ ≈ 0 . 7544 c$_{2}$ ≈ − 0 . 2456 c$_{3}$ ≈ 1 .. 3618 (17a) η ≈ 1 . 7024 β ≈ 0 . 7024 δ ≈ 2 . 4457 (17b) Fig.3(a) shows the ancient solution, while Fig.3(b) illustrates the ancient solution for a ′ 0 < FIG. 3: a ′ $^{2}$( λ ′ ) vs λ ′ for 4 th order flow 0.000 0.005 0.010 0.015 0.0 0.2 0.4 0.6 0.8 1.0 Λ ' a ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (a) a ′ 0 = 1 , λ ′ s = 0 . 019 0 20 40 60 80 100 0.5 1.0 1.5 Λ ' a ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (b) a ′ 0 = 1 , λ ′ s = − 0 . 0211 0 2 4 6 8 10 2 3 4 5 6 7 8 9 Λ ' a ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (c) a ′ 0 = 3 1 . 3048( a ′ ∞ = constant ). Fig.3(c) with a ′ 0 > 1 . 3048( a ′ ∞ = constant ) demonstrates the eternal solution. 4. 2nd order flow on H 2 Proposition III.4 2 nd order flow on hyperbolic space with α ′ > 0 generates two kinds of final metrics depending on the initial scale factors. For b ′ ( λ ′ ) 2 > 1 it is expanding and for b ′ ( λ ′ ) 2 < 1 it is converging. In both cases the scale factor asymptotically tends to 1 in backward time( b ′ −∞ = 1 ). For α ′ < 0 , we obtain an immortal solution. 9