content_image/367537.jpg content_image/367532.jpg content_image/367531.jpg content_image/367530.jpg
can be reduced to an ODE for Ω( λ ) which (after suitable rescaling) is of the form – 

where a$_{n}$ are constant coefficients. 

<img src='content_image/367537.jpg'>

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. 

• 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. 

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. 

## 1. Reduction to 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 ) 

<img src='content_image/367532.jpg'>

<img src='content_image/367531.jpg'>

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 – 

<img src='content_image/367530.jpg'>

