Nonlinear Systems Stability Analysis

$B^2\left(x\right)=4A\left(x\right)C\left(x\right)$

(2.84)

This condition may be achieved by setting A(x) and B(x) identically to zero. For A(x) ≡ 0, each coefficient, in the expansion (2.81), should be zero, or:

Terms: Coefficients:

constant (h₁₂)₀ + β₁ = 0,

x: 2(h₁₂)₁ + β₂ = 0,

x²: 3(h₁₂)₂ + β₃ = 0,

x³: 4(h₁₂)₃+β₄ = 0,

or, in general:

$\left(h_{12}\right)=\{\begin{array}{l}\frac{-{\text{β}}_{k+1}}{k+2},\text{\hspace{0.5em}\hspace{0.5em}for\hspace{0.5em}}k=0,1,2,\dots ,N,\\ 0,\text{\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}\hspace{0.5em}for\hspace{0.5em}}k>N.\end{array}$

(2.85)

Similarly, for B(x) ≡ 0,

Terms: Coefficients:

x: 2(h₁₁)₀ +2(h₁₂)₀ β₁ +2α₁ = 0,

x²: 3(h₁₁)₁ +2(h₁₂)₀ β₂ +2(h₁₂)₁β₁ + 2α₂ = 0,

x³: 4(h₁₁)₂ +2(h₁₂)₀ β₃ +2(h₁₂)₂ + 2(h₁₂)₁β₁β₁ + 2α₃ = 0,

or, in general:

${\left(h_{12}\right)}_k=\{\begin{array}{l}\frac{-1}{k+2}\left(2{\text{α}}_{k+1}+{\displaystyle \sum _{j=0}^{k}2{\left(h_{12}\right)}_j{\text{β}}_{k-j+1}}\right),\\ j=0,1,\dots ,N,\\ k-j+1\le N,\\ k=0,1,2,\mathrm{\dots },\\ 0,k>N.\end{array}$

(2.86)

Equations (2.85) and (2.86) provide an algorithm for calculating the coefficients ...