Frequency response analysis of the Bautista-Manero-Puig model with normal stress: analytical and numerical solution for large amplitudes

Abstract

We derive explicit analytical expressions for the recurrence relations using the analytical matrix method for frequency response and the Bautista-Manero-Puig model for complex fluids. The BMP model is derived from the Extended Irreversible Thermodynamics formalism and has been shown to be useful in predicting the complex rheological behavior of self-associative systems. All harmonics of the alternating normal and shear stresses in oscillatory shear with various amplitude oscillatory regimes (AOS) can be calculated analytically, i.e., small amplitude oscillatory shear (SAOS), medium amplitude oscillatory shear (MAOS), and large amplitude oscillatory shear (LAOS). We show that incorporating the effects of the first and second normal stress differences for all AOS regimes leads to the emergence of higher harmonics. We establish the limits between the different AOS regimes based on criteria suggested by the analytical method. For some typical systems, such as CTAB-NaSal, we found a satisfactory quantitative agreement with the measured behavior of AOS.

Appendices

Appendix A: Analytic matrix method for frequency response techniques of nonlinear systems (Hernandez et al. 2021)

In ref. Hernandez et al. (2021) we presented a method to obtain the analytic frequency response solution of input-state stable dynamical systems with the form

$$\begin{aligned} \dot{X}\left( t\right) =f\left( X\left( t\right) ,\zeta \left( t\right) \right) \end{aligned}$$

(A1)

where $X\subset \mathbb {R}^{n}$ is the state vector, while $\zeta \in \mathbb {R}^{2}$ is the oscillatory input vector and $f:\mathbb {R}^{n}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{n}$ is a continuous function. In particular, in traditional frequency response techniques, $\zeta$ has a linear oscillatory behavior with the following dynamics

$$\begin{aligned} \dot{\zeta }_{1}= & {} \omega \zeta _{2}\end{aligned}$$

(A2a)

$$\begin{aligned} \dot{\zeta }_{2}= & {} -\omega \zeta _{1} \end{aligned}$$

(A2b)

where $\omega$ is the frequency of oscillation. The behavior of $\zeta _{1}$ and $\zeta _{2}$ depends on the initial conditions and have the general form: $\zeta _{1}\left( t\right) =A\sin \left( \omega t+\phi \right)$ and $\zeta _{2}\left( t\right) =A\cos \left( \omega t+\phi \right)$, where A and $\phi$ are the amplitude and phase of oscillation.

After a transient period, system (A1) reaches a invariant submanifold, $\tilde{X}\left( t\right) =\xi \left( \zeta \left( t\right) \right)$, where $\tilde{\square }$ represents the variables of the oscillatory invariant submanifold, described by the invariant partial differential equation

$$\begin{aligned} \omega \left( \frac{\partial \xi \left( \zeta \right) }{\partial \zeta _{1}}\zeta _{2}-\frac{\partial \xi \left( \zeta \right) }{\partial \zeta _{2}}\zeta _{1}\right) =f\left( \xi \left( \zeta \right) ,\zeta \right) . \end{aligned}$$

(A3)

Then, the analytic frequency response solution is proposed to satisfy the following form

$$\begin{aligned} \xi _{k}\left( \zeta \right) =\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }G_{k,i,j}\zeta _{1}^{i}\zeta _{2}^{j},\qquad k=1,2,\ldots ,n \end{aligned}$$

(A4)

where $G_{k,i,j}$ are undetermined coefficients. Then, the problem reduces to find the correct value of these coefficients, that can be found using the quadrature method, where the proposed solution (A4) is substituted in Eq. (A3) to reduce the problem to find coefficients $G_{k,i,j}$. This solution can be rewritten in matrix form as

$$\begin{aligned} \xi _{k}\left( \zeta \right) =Z_{1}^{T}G_{k}Z_{2} \end{aligned}$$

(A5)

where

$$\begin{aligned} Z_{i}=\begin{pmatrix}1\\ \zeta _{i}\\ \zeta _{i}^{2}\\ \zeta _{i}^{3}\\ \vdots \end{pmatrix},\quad \text {for}\quad i=1,2, \end{aligned}$$

(A6)

and

$$\begin{aligned} G_{k}=\begin{pmatrix}G_{k,0,0} &{} G_{k,0,1} &{} G_{k,0,2} &{} \cdots \\ G_{k,1,0} &{} G_{k,1,1} &{} G_{k,1,2} &{} \cdots \\ G_{k,2,0} &{} G_{k,2,1} &{} G_{k,2,2} &{} \cdots \\ \vdots &{} \vdots &{} \vdots &{} \ddots \end{pmatrix}. \end{aligned}$$

(A7)

The following two definitions are instrumental to develop the solution method.

Definition 1

Given two square matrices $\Phi _{a}$ and $\Phi _{b}$ with the same dimension $N\times N$ (N may tend to infinity), the product operator is defined as

$$\begin{aligned} \left\langle \Phi _{a},\Phi _{b}\right\rangle _{N}\!=\!\sum _{k\!=\!0}^{N-1}\left[ \begin{pmatrix}\Phi _{a}\mathcal {E}_{k}&\!\!\mathcal {P}^{T}\Phi _{a}\mathcal {E}_{k}&\!\!\left( \mathcal {P}^{2}\right) ^{T}\!\Phi _{a}\mathcal {E}_{k}&\!\!\cdots&\left( \mathcal {P}^{N-1}\right) ^{T}\Phi _{a}\mathcal {E}_{k}\end{pmatrix}\Phi _{b}\mathcal {P}^{k}\right] \end{aligned}$$

(A8)

where the elements of matrix $\mathcal {P}$ are

$$\begin{aligned} \left[ \mathcal {P}\right] _{i,j}={\left\{ \begin{array}{ll} 1 &{} \text {if }j=i+1\\ 0 &{} \text {otherwise} \end{array}\right. }, \end{aligned}$$

(A9)

while $\mathcal {E}$ is a vector of dimension $N\times 1$, whose elements are

$$\begin{aligned} \left[ \mathcal {E}_{k}\right] _{i,1}={\left\{ \begin{array}{ll} 1 &{} \text {if }i=k+1\\ 0 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

(A10)

Definition 2

Given the square matrix $\Phi _{a}$ of dimension $N\times N$, the differential operator is defined as

$$\begin{aligned} d\Phi _{a}=\mathcal {D}^{T}\Phi _{a}\mathcal {P}-\mathcal {P}^{T}\Phi _{a}\mathcal {D} \end{aligned}$$

(A11)

where the elements of matrix $\mathcal {D}$ are

$$\begin{aligned} \left[ \mathcal {D}\right] _{i,j}={\left\{ \begin{array}{ll} j &{} \text {if }i=j+1\\ 0 &{} \text {otherwise} \end{array}\right. }, \end{aligned}$$

(A12)

while the elements of matrix $\mathcal {P}$ are defined in Eq. (A9).

In Hernandez et al. (2021) we proved that the time derivative of $\xi _{k}\left( \zeta \right)$ is

$$\begin{aligned} \frac{d\xi _{k}\left( \zeta \right) }{dt}=\frac{d}{dt}\left( Z_{1}^{T}G_{k}Z_{2}\right) =\omega Z_{1}^{T}dG_{k}Z_{2} \end{aligned}$$

(A13)

where, taking the structure presented in Definition 2, it hold that $dG_{k}=\mathcal {D}^{T}G_{k}\mathcal {P}-\mathcal {P}^{T}G_{k}\mathcal {D}$, while the product of the functions $\phi _{a}\left( z_{1},z_{2}\right) =Z_{1}^{T}\Phi _{a}Z_{2}$ and $\phi _{b}\left( z_{1},z_{2}\right) =Z_{1}^{T}\Phi _{b}Z_{2}$ can be expressed as

$$\begin{aligned} \phi _{a}\left( z_{1},z_{2}\right) \phi _{b}\left( z_{1},z_{2}\right)= & {} \left( Z_{1}^{T}\Phi _{a}Z_{2}\right) \left( Z_{1}^{T}\Phi _{b}Z_{2}\right) \nonumber \\= & {} Z_{1}^{T}\left\langle \Phi _{a},\Phi _{b}\right\rangle _{\infty }Z_{2} \end{aligned}$$

(A14)

where $\left\langle \Phi _{a},\Phi _{b}\right\rangle _{\infty }$ is given in Definition 1.

In Hernandez et al. (2021) it was also shown that, if each element of function $f\left( \xi \left( \zeta \right) ,\zeta \right)$ in Eq. (A3) can be expressed as series of integer powers of the elements of $\xi$ and $\zeta$, then, using the property expressed in Eq. (A14) it holds that $f_{k}\left( \xi \left( \zeta \right) ,\zeta \right) =Z_{1}^{T}F_{k}\left( G_{1},G_{2},\ldots ,G_{n}\right) Z_{2}$ and considering the time derivative (A13), any set of partial differential equations of the form (A3) can be transformed to

$$\begin{aligned} 0=Z_{1}^{T}\left[ \omega dG_{k}-F_{k}\left( G_{1},G_{2},\ldots ,G_{n}\right) \right] Z_{2},\qquad k=1,2,\ldots ,n \end{aligned}$$

(A15)

where $F_{k}\left( G_{1},G_{2},\ldots ,G_{n}\right)$ is a matrix that depends on the values of coefficients matrices $G_{1},G_{2},\ldots ,G_{n}$ defined in Eq. (A7). Independently of the values of vectors $Z_{1}$ and $Z_{2}$, Eq. (A15) holds if

$$\begin{aligned} \omega dG_{k}-F_{k}\left( G_{1},G_{2},\ldots ,G_{n}\right) =0,\qquad k=1,2,\ldots ,n, \end{aligned}$$

(A16)

then the solution of oscillatory invariant submanifold reduces to find coefficients $G_{k,i,j}$, $k=1,2,\ldots ,n$, $i,j\in \mathbb {N}$, by solving Eq. (A16).

Appendix B: Procedure of the analytic solution

By replacing Eq. (15) in the oscillatory invariant submanifold described in Eq. (13a13b13c13d) it holds that

$$\begin{aligned} 0= & {} Z_{1}^{T}\left[ \omega \left( dG_{1}-W_{5}-B_{0,1}\right) +G_{1}+W_{1}\right] Z_{2}\end{aligned}$$

(B17a)

$$\begin{aligned} 0= & {} Z_{1}^{T}\left[ \omega \left( dG_{2}-2W_{4}\right) +G_{2}+W_{2}\right] Z_{2}\end{aligned}$$

(B17b)

$$\begin{aligned} 0= & {} Z_{1}^{T}\left[ \omega dG_{3}+G_{3}+W_{3}-\psi \omega ^{2}B_{0,2}\right] Z_{2}\end{aligned}$$

(B17c)

$$\begin{aligned} 0\!=\! & {} Z_{1}^{T}\left[ \frac{dG_{4}\!+\!\frac{1}{\lambda \omega }G_{4}}{\kappa _{0}}-\phi _{\infty }W_{4}+W_{6}+\vartheta \omega ^{2}\left( W_{8}-\phi _{\infty }W_{7}\right) \right] Z_{2} \end{aligned}$$

(B17d)

where

$$ dG_{k}=\mathcal {D}^{T}G_{k}\mathcal {P}-\mathcal {P}^{T}G_{k}\mathcal {D},\qquad \text {for }k=1,2,3,4 $$

is the differential operator described in Eq. (A11) of Definition 2, while with the aid of the general form of two functions product, given in Eq. (A14), the following expressions have been derived:

$$\begin{aligned} \xi _{i}\xi _{4}= & {} \left( Z_{1}^{T}G_{i}Z_{2}\right) \left( Z_{1}^{T}G_{4}Z_{2}\right) =Z_{1}^{T}W_{i}Z_{2}\quad \text {for }i=1,2,3,\\ \xi _{1}\zeta _{2}= & {} \left( Z_{1}^{T}G_{1}Z_{2}\right) \left( Z_{1}^{T}B_{0,1}Z_{2}\right) =Z_{1}^{T}W_{4}Z_{2},\\ \xi _{3}\zeta _{2}= & {} \left( Z_{1}^{T}G_{3}Z_{2}\right) \left( Z_{1}^{T}B_{0,1}Z_{2}\right) =Z_{1}^{T}W_{5}Z_{2},\\ \zeta _{2}^{2}= & {} Z_{1}^{T}B_{0,2}Z_{2},\\ \xi _{1}\xi _{4}\zeta _{2}= & {} \left( Z_{1}^{T}W_{1}Z_{2}\right) \left( Z_{1}^{T}B_{0,1}Z_{2}\right) =Z_{1}^{T}W_{6}Z_{2},\\ \xi _{1}\zeta _{2}^{3}= & {} \left( Z_{1}^{T}W_{4}Z_{2}\right) \left( Z_{1}^{T}B_{0,2}Z_{2}\right) =Z_{1}^{T}W_{7}Z_{2},\quad \text {and}\\ \xi _{1}\xi _{4}\zeta _{2}^{3}= & {} \left( Z_{1}^{T}W_{6}Z_{2}\right) \left( Z_{1}^{T}B_{0,2}Z_{2}\right) =Z_{1}^{T}W_{8}Z_{2}, \end{aligned}$$

where matrices $W_{k}$ for $k=1,2,\ldots ,8$ are defined in Eq. (17a17b17c17d17e17f).

According to Definitions of Appendix A, the element of the mth row and nth $dG_{k}$ are²

$$\begin{aligned} \left[ dG_{k}\right] _{m,n}={\left\{ \begin{array}{ll} 0 &{} m,n=0\\ -\left( n+1\right) G_{k,m-1,n+1} &{} n=0,m=1,2,3,\ldots \\ \left( m+1\right) G_{k,m+1,n-1} &{} m=0,n=1,2,3,\ldots \\ -\left( n+1\right) G_{k,m-1,n+1} &{} m,n=1,2,3,\ldots \\ \quad +\left( m+1\right) G_{k,m+1,n-1}&{}\\ \end{array}\right. } \end{aligned}$$

(B18)

while the element of the mth row and nth column of matrices $W_{k}$ are

$$\begin{aligned} \left[ W_{k}\right] _{m,n}= & {} \sum _{i=0}^{m}\sum _{j=0}^{n}G_{k,i,n-j}G_{4,m-i,j},\qquad k=1,2,3.\end{aligned}$$

(B19a)

$$\begin{aligned} \left[ W_{4}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} 0 &{} n=0\\ G_{1,m,n-1} &{} n=1,2,3,\ldots \end{array}\right. }\end{aligned}$$

(B19b)

$$\begin{aligned} \left[ W_{5}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} 0 &{} n=0\\ G_{3,m,n-1} &{} n=1,2,3,\ldots \end{array}\right. }\end{aligned}$$

(B19c)

$$\begin{aligned} \left[ W_{6}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} 0 &{} n=0\\ \sum _{i=0}^{m}\sum _{j=0}^{n-1}G_{1,i,n-1-j}G_{4,m-i,j} &{} n=1,2,3,\ldots \end{array}\right. }\end{aligned}$$

(B19d)

$$\begin{aligned} \left[ W_{7}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} 0 &{} n=0,1,2\\ G_{1,m,n-3} &{} n=3,4,5,\ldots \end{array}\right. } \end{aligned}$$

(B19e)

$$\begin{aligned} \left[ W_{8}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} 0 &{} n=0,1,2\\ \sum _{i=0}^{m}\sum _{j=0}^{n-3}G_{1,i,n-3-j}G_{4,m-i,j} &{} n=3,4,5,\ldots \end{array}\right. } \end{aligned}$$

(B19f)

Equation (B17aB17bB17cB17d) hold for any vectors $Z_{1}$ and $Z_{2}$ if

$$\begin{aligned} \omega dG_{1}+G_{1}= & {} \omega \left( W_{5}+B_{0,1}\right) -W_{1}=:H_{1}\end{aligned}$$

(B20a)

$$\begin{aligned} \omega dG_{2}+G_{2}= & {} 2\omega W_{4}-W_{2}=:H_{2}\end{aligned}$$

(B20b)

$$\begin{aligned} \omega dG_{3}+G_{3}= & {} -W_{3}+\psi \omega ^{2}B_{0,2}=:H_{3}\end{aligned}$$

(B20c)

$$\begin{aligned} \lambda \omega dG_{4}+G_{4}= & {} \kappa _{0}\lambda \omega \left[ \phi _{\infty }W_{4}-W_{6}\right. \nonumber \\{} & {} \left. +\vartheta \omega ^{2}\left( \phi _{\infty }W_{7}-W_{8}\right) \right] =:H_{4} \end{aligned}$$

(B20d)

hold. Notice that Eq. (B20aB20bB20cB20d) is equivalent to Eq. (16a16b16c16d). Given definition (B18), the element of the mth row and nth column of matrices in the left-hand side of Eq. (B20aB20bB20cB20d) are

$$\begin{aligned} \left[ \omega dG_{k}+G_{k}\right] _{m,n}={\left\{ \begin{array}{ll} G_{k,0,0} &{} m,n=0\\ G_{k,m,n}-\left( n+1\right) &{} n=0,m=1,2,3,\ldots \\ \quad \omega G_{k,m-1,n+1}&{}\\ \left( m+1\right) \omega G_{k,m+1,n-1} &{} m=0,n=1,2,3,\ldots \\ \quad +G_{k,m,n}&{}\\ \left( m+1\right) \omega G_{k,m+1,n-1}\\ +G_{k,m,n}-\left( n+1\right) &{} m,n=1,2,3,\ldots \\ \quad \omega G_{k,m-1,n+1}&{}\\ \end{array}\right. } \end{aligned}$$

(B21)

for $k=1,2,3$ and

$$\begin{aligned} \left[ \lambda \omega dG_{4}+G_{4}\right] _{m,n}={\left\{ \begin{array}{ll} G_{4,0,0} &{} m,n=0\\ G_{4,m,n}-\left( n+1\right) \lambda \omega &{} n=0,m=1,2,3,\ldots \\ \quad G_{4,m-1,n+1}&{}\\ \left( m+1\right) &{} m=0,n=1,2,3,\ldots \\ \quad \lambda \omega G_{4,m+1,n-1}+G_{4,m,n} &{}\\ \left( m+1\right) \lambda \omega G_{4,m+1,n-1}\\ +G_{4,m,n}-\left( n+1\right) &{} m,n=1,2,3,\ldots \\ \omega G_{4,m-1,n+1}&{}\\ \end{array}\right. } \end{aligned}$$

(B22)

respectively, and considering definition (B19aB19bB19cB19dB19e), the element of the mth row and nth column of matrices in the right-hand side of Eq. (B20aB20bB20cB20d) are

$$\begin{aligned} \left[ H_{1}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} -\sum _{i=0}^{m}G_{1,i,0}G_{4,m-i,0}, &{} m=0,1,2,\ldots ,n=0\\ \omega -\sum _{i=0}^{1}G_{1,i,0}G_{4,1-i,0}, &{} m=0,n=1\\ \omega G_{3,m,n-1}\\ -\sum _{i=0}^{m}\sum _{j=0}^{n}G_{1,i,n-j}G_{4,m-i,j}, &{} \text {otherwise} \end{array}\right. }\end{aligned}$$

(B23a)

$$\begin{aligned} \left[ H_{2}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} -\sum _{i=0}^{m}G_{2,i,0}G_{4,m-i,0} &{} m=0,1,2,\ldots ,n=0\\ 2\omega G_{1,m,n-1}\\ -\sum _{i=0}^{m}\sum _{j=0}^{n}G_{2,i,n-j}G_{4,m-i,j} &{} \text {otherwise} \end{array}\right. }\end{aligned}$$

(B23b)

$$\begin{aligned} \left[ H_{3}\right] _{m,n}= & {} {\left\{ \begin{array}{ll} \psi \omega ^{2}-\sum _{i=0}^{2}G_{3,i,0}G_{4,2-i,0} &{} m=0,n=2\\ -\sum _{i=0}^{m}\sum _{j=0}^{n}G_{3,i,n-j}G_{4,m-i,j} &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

(B23c)

$$\begin{aligned} \frac{\left[ H_{4}\right] _{m,n}}{\kappa _{0}\lambda \omega }= & {} {\left\{ \begin{array}{ll} 0 &{} m=0,1,2,\ldots ,n=0\\ \phi _{\infty }G_{1,m,n-1}\\ -\sum _{i=0}^{m}\sum _{j=0}^{n-1}G_{1,i,n-1-j} &{} m=0,1,2,\ldots ,n=1,2\\ \quad G_{4,m-i,j}&{}\\ \phi _{\infty }\left( G_{1,m,n-1}+\vartheta \omega ^{2}G_{1,m,n-3}\right) \\ -\sum _{i=0}^{m}\sum _{j=0}^{n-1}G_{1,i,n-1-j}\\ \quad G_{4,m-i,j}\\ -\vartheta \omega ^{2}\sum _{i=0}^{m}\sum _{j=0}^{n-3}&{} \text {otherwise}\\ \quad G_{1,i,n-3-j}G_{4,m-i,j}&{} \end{array}\right. } \end{aligned}$$

(B23d)

Taking into account these expressions, the elements of order zero, i.e., the elements of the first row and first column $\left( m=0,n=0\right)$ of Eq. (B20aB20bB20cB20d), are

$$\begin{aligned} \left[ H_{1}\right] _{0,0}= & {} G_{1,0,0}=-G_{1,0,0}G_{4,0,0}\\ \left[ H_{2}\right] _{0,0}= & {} G_{2,0,0}=-G_{2,0,0}G_{4,0,0}\\ \left[ H_{2}\right] _{0,0}= & {} G_{3,0,0}=-G_{3,0,0}G_{4,0,0}\\ \left[ H_{2}\right] _{0,0}= & {} G_{4,0,0}=0 \end{aligned}$$

It follows that $G_{k,0,0}=0$ for $k=1,2,3,4$. Replacing these values, the elements of order one, i.e., the elements of the second row and first column $\left( m=1,n=0\right)$ together with the first row and second column $\left( m=0,n=1\right)$ of Eq. (B20aB20bB20cB20d) produce the linear systems of equations

$$\begin{aligned} \begin{pmatrix}\left[ H_{1}\right] _{1,0}\\ \left[ H_{1}\right] _{0,1} \end{pmatrix}=\begin{pmatrix}1 &{} -\omega \\ \omega &{} 1 \end{pmatrix}\begin{pmatrix}G_{1,1,0}\\ G_{1,0,1} \end{pmatrix}=\begin{pmatrix}0\\ \omega \end{pmatrix} \end{aligned}$$

(B24a)

and

$$\begin{aligned} \begin{pmatrix}\left[ H_{k}\right] _{1,0}\\ \left[ H_{k}\right] _{0,1} \end{pmatrix}=\begin{pmatrix}1 &{} -\omega \\ \omega &{} 1 \end{pmatrix}\begin{pmatrix}G_{k,1,0}\\ G_{k,0,1} \end{pmatrix}=\begin{pmatrix}0\\ 0 \end{pmatrix},\qquad \text {for }k=2,3,4. \end{aligned}$$

(B24b)

that have the solutions $G_{1,1,0}=\frac{\omega ^{2}}{\omega ^{2}+1}$, $G_{1,0,1}=\frac{\omega }{\omega ^{2}+1}$, and $G_{k,1,0}=G_{k,0,1}=0$ for $k=2,3,4$. Following the same procedure, the elements of order two, i.e., the elements of the third row and first column $\left( m=2,n=0\right)$ together with the second row and second column $\left( m=1,n=1\right)$ and the first row and third column $\left( m=0,n=2\right)$ of Eq. (B20aB20bB20cB20d) produce the linear systems of equations

$$\begin{aligned} \begin{pmatrix}\left[ H_{1}\right] _{2,0}\\ \left[ H_{1}\right] _{1,1}\\ \left[ H_{1}\right] _{0,2} \end{pmatrix}= & {} \begin{pmatrix}1 &{} -\omega &{} 0\\ 2\omega &{} 1 &{} -2\omega \\ 0 &{} \omega &{} 1 \end{pmatrix}\begin{pmatrix}G_{1,2,0}\\ G_{1,1,1}\\ G_{1,0,2} \end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0 \end{pmatrix}\end{aligned}$$

(B25a)

$$\begin{aligned} \begin{pmatrix}\left[ H_{2}\right] _{2,0}\\ \left[ H_{2}\right] _{1,1}\\ \left[ H_{2}\right] _{0,2} \end{pmatrix}= & {} \begin{pmatrix}1 &{} -\omega &{} 0\\ 2\omega &{} 1 &{} -2\omega \\ 0 &{} \omega &{} 1 \end{pmatrix}\begin{pmatrix}G_{2,2,0}\\ G_{2,1,1}\\ G_{2,0,2} \end{pmatrix}=\begin{pmatrix}0\\ 2\omega G_{1,1,0}\\ 2\omega G_{1,0,1} \end{pmatrix}\end{aligned}$$

(B25b)

$$\begin{aligned} \begin{pmatrix}\left[ H_{3}\right] _{2,0}\\ \left[ H_{3}\right] _{1,1}\\ \left[ H_{3}\right] _{0,2} \end{pmatrix}= & {} \begin{pmatrix}1 &{} -\omega &{} 0\\ 2\omega &{} 1 &{} -2\omega \\ 0 &{} \omega &{} 1 \end{pmatrix} \begin{pmatrix}G_{3,2,0}\\ G_{3,1,1}\\ G_{3,0,2} \end{pmatrix}=\begin{pmatrix}0\\ 0\\ \psi \omega ^{2} \end{pmatrix} \end{aligned}$$

(B25c)

$$\begin{aligned} \begin{pmatrix}\left[ H_{4}\right] _{2,0}\\ \left[ H_{4}\right] _{1,1}\\ \left[ H_{4}\right] _{0,2} \end{pmatrix}= & {} \begin{pmatrix}1 &{} -\lambda \omega &{} 0\\ 2\lambda \omega &{} 1 &{} -2\lambda \omega \\ 0 &{} \lambda \omega &{} 1 \end{pmatrix}\begin{pmatrix}G_{4,2,0}\\ G_{4,1,1}\\ G_{4,0,2} \end{pmatrix}\nonumber \\= & {} \begin{pmatrix}0\\ \kappa \lambda \omega \phi _{\infty }G_{1,1,0}\\ \kappa \lambda \omega \phi _{\infty }G_{1,0,1} \end{pmatrix} \end{aligned}$$

(B25d)

with solution $G_{1,2,0}=G_{1,1,1}=G_{1,0,2}=0$,

$$\begin{aligned} \begin{pmatrix}G_{2,2,0}\\ G_{2,1,1}\\ G_{2,0,2} \end{pmatrix}= & {} \frac{2\omega ^{2}}{\left( 1+\omega ^{2}\right) \left( 1+4\omega ^{2}\right) }\begin{pmatrix}3\omega ^{2}\\ 3\omega \\ 1+\omega ^{2} \end{pmatrix},\\ \begin{pmatrix}G_{3,2,0}\\ G_{3,1,1}\\ G_{3,0,2} \end{pmatrix}= & {} \frac{\omega ^{2}\psi }{1+4\omega ^{2}}\begin{pmatrix}2\omega ^{2}\\ 2\omega \\ 1+2\omega ^{2} \end{pmatrix},\qquad \qquad \qquad \text {and}\\ \begin{pmatrix}G_{4,2,0}\\ G_{4,1,1}\\ G_{4,0,2} \end{pmatrix}= & {} \frac{\kappa \lambda \phi _{\infty }\omega ^{2}}{\left( 1+\omega ^{2}\right) \left( 1+4\lambda ^{2}\omega ^{2}\right) }\begin{pmatrix}\left( 1+2\lambda \right) \lambda \omega ^{2}\\ \left( 1+2\lambda \right) \omega \\ 1+\left( 2\lambda -1\right) \lambda \omega ^{2} \end{pmatrix}. \end{aligned}$$

Following this procedure, the vectors of coefficients for the terms of order n are defined as:

$$ G_{i}^{n}=\begin{pmatrix}G_{i,n,0}\\ G_{i,n-1,1}\\ \vdots \\ G_{i,0,n} \end{pmatrix},\qquad \text {for }i=1,2,3,4. $$

Then, for n odd, with $n\ge 3$, it holds that $G_{2}^{n}=G_{3}^{n}=G_{4}^{n}=0$, while $G_{1}^{n}$ is the solution of the system of equations:

$$\begin{aligned} \left( I_{n+1}+\omega B_{n+1}\right) G_{1}^{n}=C_{n+1}\qquad \text {for }n\text { odd} \end{aligned}$$

(B26)

where $I_{n+1}$ is the identity matrix of dimension $n+1$, the elements of matrix $B_{n+1}$ are

$$ \left[ B_{n+1}\right] _{i,j}={\left\{ \begin{array}{ll} -i &{} j=i+1,i=1,2,\ldots ,n,\\ \left( n-j\right) &{} i=j+1,j=1,2,\ldots ,n,\\ 0 &{} \text {otherwise}, \end{array}\right. } $$

while the elements of vector $C_{n+1}$ are

$$\begin{aligned} \left[ C_{n+1}\right] _{1}= & {} -\sum _{j=0}^{n}G_{1,j,0}G_{4,n-j,0}\\ \left[ C_{n+1}\right] _{k}= & {} \omega G_{3,n-k+1,k-2}\\{} & {} -\sum _{i=0}^{k-1}\sum _{j=0}^{n-k+1}G_{1,j,k-i-1}G_{4,n-k-j+1,i},\quad k=2,3,\ldots ,n+1 \end{aligned}$$

Notice that Eq. (B26) is linear with respect to $G_{1}^{n}$ while $C_{n+1}$ has nonlinear terms (products of $G_{1,j,k-i-1}G_{4,n-k-j+1,i}$) that are coefficients with smaller subindex that those previously computed.

For n even, with $n\ge 4$, it holds that $G_{1}^{n}=0$, while $G_{2}^{n}$, $G_{3}^{n}$, and $G_{4}^{n}$ are the solution of the following system of equations:

$$\begin{aligned} \left( I_{n+1}+\omega B_{n+1}\right) G_{2}^{n}= & {} D_{n+1}\qquad \text {for }n\text { even}\end{aligned}$$

(B27)

$$\begin{aligned} \left( I_{n+1}+\omega B_{n+1}\right) G_{3}^{n}= & {} E_{n+1}\qquad \text {for }n\text { even}\end{aligned}$$

(B28)

$$\begin{aligned} \left( I_{n+1}+\lambda \omega B_{n+1}\right) G_{4}^{n}= & {} \kappa \lambda \omega F_{n+1}\qquad \text {for }n\text { even} \end{aligned}$$

(B29)

where

$$\begin{aligned} \left[ D_{n+1}\right] _{1}= & {} -\sum _{j=0}^{n}G_{2,j,0}G_{4,n-j,0}\\ \left[ D_{n+1}\right] _{k}= & {} 2\omega G_{1,n-k+1,k-2}\\{} & {} -\sum _{i=0}^{k-1}\sum _{j=0}^{n-k+1}G_{2,j,k-i-1}G_{4,n-k-j+1,i},\qquad k=2,3,\ldots ,n+1 \end{aligned}$$

$$ \left[ E_{n+1}\right] _{k}=-\sum _{i=0}^{k-1}\sum _{j=0}^{n-k+1}G_{3,j,k-i-1}G_{4,n-k-j+1,i},\qquad k=2,3,\ldots ,n+1 $$

$$\begin{aligned} \left[ F_{n+1}\right] _{1}= & {} 0\\ \left[ F_{n+1}\right] _{2}= & {} \phi _{\infty }G_{1,n-1,0}-\sum _{j=0}^{n-1}G_{1,n-j-1,0}G_{4,j,0}\\ \left[ F_{n+1}\right] _{3}= & {} \phi _{\infty }G_{1,n-2,0}-\sum _{i=0}^{1}\sum _{j=0}^{n-2}G_{1,n-j-2,i}G_{4,j,1-i}\\ \left[ F_{n+1}\right] _{k}= & {} \phi _{\infty }\left( G_{1,n-k+1,k-2}+\vartheta \omega ^{2}G_{1,n-k+1,k-4}\right) \\{} & {} -\sum _{i=0}^{k-2}\sum _{j=0}^{n-k+1}G_{1,n-k-j+1,i}G_{4,j,k-i-2}\\{} & {} -\vartheta \omega ^{2}\sum _{i=0}^{k-4}\sum _{j=0}^{n-k+1}G_{1,n-k-j+1,i}G_{4,j,k-i-4},\quad k=4,5,\ldots ,n+1 \end{aligned}$$

Notice that similarly to the case of n odd, Eqs. (B27)–(B29) are linear with respect to $G_{2}^{n}$, $G_{3}^{n}$, and $G_{4}^{n}$, respectively, while $D_{n+1}$, $E_{n+1}$, and $F_{n+1}$ have nonlinear terms that are coefficients with smaller subindex than those previously computed.