Theory and Modern Applications
Table 1 The first twist coefficient for some values of \(a,b,c \geq 0 \)
From: Stability analysis of a certain class of difference equations by using KAM theory
Parameters | Equation | The first twist coefficient |
|---|---|---|
a = b = 0, c>1, x̄ = c − 1 | \(y_{n+1}=\frac{cy_{n}^{2}}{(1+y_{n})y_{n-1}}\) | \(\alpha _{1}=\frac{c}{12 c^{2}+10 c+2}\) |
a = c = 0, \(\bar{x}=\frac{1}{2} (\sqrt{4 b+1}-1 )\) | \(y_{n+1}=\frac{by_{n}}{(1+y_{n})y_{n-1}}\) | \(\alpha _{1}=\frac{(b-1) b}{2 (4 b+1) (3 (\sqrt{4 b+1}+1 )+b (\sqrt{4 b+1}+6 ) )}\) |
b = c = 0, \(a=\bar{x}^{3}+\bar{x}^{2}\) | \(y_{n+1}=\frac{a}{(1+y_{n})y_{n-1}}\) | \(\alpha _{1}=\frac{ (\bar{x}-1 ) \bar{x}}{2 (\bar{x}+2 ) (3 \bar{x}+2 )^{2}}\) |
c = 0, \(a=\bar{x}^{3}+\bar{x}^{2}-b\bar{x}\) | \(y_{n+1}=\frac{a+b y_{n}}{(1+y_{n})y_{n-1}}\) | \(\alpha _{1}=\frac{ (b-\bar{x}^{2} ) (2 b^{3} \bar{x}-4 b \bar{x}^{4}-(b (b+5)-1) \bar{x}^{3}+(b-2) b \bar{x}^{2}-\bar{x}^{5}+b^{3} )}{2 (\bar{x}+1 ) (\bar{x}+b ) (\bar{x} (\bar{x}+2 )+b ) (b-\bar{x} (3 \bar{x}+2 ) )^{2}}\) |