0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 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 . 0665 0.00 0.05 0.10 0.15 0.0 0.2 0.4 0.6 0.8 1.0 Λ ' a ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (b) a ′ 0 = 1 , λ ′ s = 0 . 152 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.5 1.0 1.5 2.0 Λ ' a ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (c) a ′ 0 = 2 , λ ′ s = 0 . 288 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1.0 1.5 2.0 Λ ' a ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (d) a ′ 0 = 2 , λ ′ s = 0 . 696 FIG. 2: a ′ $^{2}$( λ ′ ) vs λ ′ for 3 rd order flow 3. 4th order flow on S 2 Proposition III.3 Inclusion of the 4 th order terms in the flow on a canonical two-sphere, we observe the following behavior: • α ′ > 0 and any choice of a ′ 0 gives rise to ancient solution. • α ′ < 0 , a ′ 0 = 1 . 3048 : soliton. For a ′ 0 = 1 . 3048 we find, a ′ ∞ =const • α ′ < 0; a ′ 0 < 1 . 3048 : immortal solution and . a ′ 0 > 1 . 3048 : eternal solution. The solution of the flow equation with the fourth order term included is found to be, where the constants are: 8 C = a ′ 2 0 + c$_{1}$ ln[ a ′ 2 0 − η ] − c$_{2}$ 2 ln [ a ′ 4 0 + βa ′ 2 0 + δ ] − 2 c$_{3}$ − βc$_{2}$ 2 √ δ − β 2 4 tan − 1   a ′ 2 0 + β 2 √ δ − β 2 4   (16) a ′ 2 + c$_{1}$ ln[ a ′ 2 − η ] − c$_{2}$ 2 ln [ a ′ 4 + βa ′ 2 + δ ] − 2 c$_{3}$ − βc$_{2}$ 2 √ δ − β 2 4 tan − 1   a ′ 2 + β 2 √ δ − β 2 4   = − 2 λ ′ + C (15)