can be reduced to an ODE for Ω( λ ) which (after suitable rescaling) is of the form – where a$_{n}$ are constant coefficients. Proposition IV.3 Given Ω( λ = 0) = Ω$_{0}$ ≥ 0 , the ODE Eq.(27) has a unique expanding solution ( d Ω dλ > 0 ), with a finite time singularity in the past at λ = λ$_{s}$ ≤ 0 , for each N = 1 , 3 (odd) and also for N = 2 , 4 (even) for α ′ < 0 , Proposition IV.4 For N = even with α ′ > 0 , Eq.(27) has 3 distinct solutions. with ξ$_{2}$ = 1 and ξ$_{4}$ ≈ 1 . 5636 . Next we prove these propositions by reducing the flow Eq.(1) to an ODE of the form Eq.(27) and then explicitly solving the ODE. Using the metric of Eq.(25) in the flow Eq.(7) we get the following dynamical equations for f and r (up to 2 nd order ) where ˙ F = ∂F ∂λ and F ′ = ∂F $_{∂σ}$, for any function F . To solve the above equations we assume that f and r can be written in a variable separable form as – 16 f ( σ, λ ) = f$_{σ}$ ( σ ) + f$_{λ}$ ( λ ) ; r ( σ, λ ) = r$_{λ}$ ( λ ) r$_{σ}$ ( σ ) (30) ˙ r = 4 r [ f ′′ + f ′ 2 − f ′ r ′ r ] − 4 α ′ r 3 [ f ′′ + f ′ 2 − f ′ r ′ r ] 2 (29) ˙ f = 1 r 2 [( f ′′ + f ′ 2 − f ′ r ′ r ) + 3 f ′ 2 ] − α ′ r 4 [ ( f ′′ + f ′ 2 − f ′ r ′ r ) 2 + 3 f ′ 4 ] (28) 1. Reduction to ODE • Ω > α ′ k $^{2}$ξ$_{N}$ : An eternal expanding solution without finite time singularities. • Ω < α ′ k $^{2}$ξ$_{N}$ : A contracting solution ending in a singularity at finite λ = λ$_{s}$ ≥ 0 . • Ω = α ′ k $^{2}$ξ$_{N}$ : A fixed point soliton solution. d Ω dλ = $^{N}$− 1 ∑ n =0 a$_{n}$ Ω − n (27)