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Fig.3(a) shows the ancient solution, while Fig.3(b) illustrates the ancient solution for a ′ 0 < 

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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. 

