Codimension-one and codimension-two bifurcations in a new discrete chaotic map based on gene regulatory network model

Abstract

In this paper, the stability and bifurcations of a new discrete chaotic map based on gene regulatory network are studied. Firstly, the existence and stability conditions of the fixed points are given. Secondly, the conditions for existence of three cases of codimension-one bifurcations (fold bifurcation, flip bifurcation and Neimark–Sacker bifurcation) are derived by using the center manifold theorem and bifurcation theory. Then, the conditions for the occurrence of codimension-two bifurcation (fold–flip bifurcation, 1:2, 1:3 and 1:4 strong resonance) are investigated by using several variable substitutions and introduction of new parameters. Meanwhile, these bifurcation curves are returned to the original variables and parameters to express for easy verification. The corresponding numerical simulations and numerical continuation results not only show the validity of the proposed results, but also exhibit the interesting and complex dynamical behaviors. Finally, some initial conditions and two-parameter spaces analysis are given numerically. The local attraction basins and two-parameter space plots display interesting dynamical behaviors of the discrete system operating with different integral step size and other parameters changing.

Appendices

Appendix A: $a_{ij}$ and $F_i(X_1,L_2,Y_1)$ in (3.4); $e_i$ in (3.9); $b_i$ in (3.10)

$$\begin{aligned}&a_{11} = 1 + \delta (k_1A - l_1),~~ a_{13} = -\delta k_1B,~~ a_{31} = \delta k_2A,\\&a_{32} = -\delta y^*,~~ a_{33} = 1 - \delta (k_2B + l_2^*),\\&F_1(X_1,L_2,Y_1) = f_{200}X_1^2 + f_{101}X_1Y_1 \\&\qquad + f_{002}Y_1^2 +\mathcal {O}\left( \Vert (X_1,L_2,Y_1)\Vert ^3\right) ,\\&F_2(X_1,L_2,Y_1) = g_{200}X_1^2 + g_{101}X_1Y_1 + g_{011}L_2Y_1 + g_{002}Y_1^2 \\&\qquad + \mathcal {O}\left( \Vert (X_1,L_2,Y_1)\Vert ^3\right) ,\\&f_{200} \!=\! \frac{k_1pq\left( -3q^2x^{*2}\! -\! 3qx^{*2}y^* \!+\! pq^2 \!+\! 2pqy^* \!+\! py^{*2}\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ,\\&f_{101} = \frac{2k_1pqx^*\left( -qx^{*2} + pq + py^*\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ,\\&f_{002} = \frac{k_1qp^2x^{*2}}{\left( qx^{*2} + pq + py^*\right) ^3}\delta ,\\&g_{200} = \frac{k_2pq\left( -3q^2x^{*2}\! -\! 3qx^{*2}y^* \!+\! pq^2 \!+\! 2pqy^* \!+\! py^{*2}\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ,\\&g_{101} = \frac{2k_2pqx^*\left( -qx^{*2} \!+\! pq \!+\! py^*\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ,~~ g_{011} = -\delta ,\\&g_{002} = \frac{k_2qp^2x^{*2}}{\left( qx^{*2} + pq + py^*\right) ^3}\delta .\\&e_1 = a_{13}^2g_{200} + a_{13}(-a_{11} + 1)g_{101} + (a_{11} - 1)^2g_{002},\\&e_2 = -a_{13}a_{32}(a_{11}k_1 + a_{13}k_2 - k_1)g_{101} \\&\quad - a_{13}k_2(\lambda _{12} - 1)(a_{11} - 1)g_{011} \\&\quad + 2a_{32}(a_{11} - 1)(a_{11}k_1 + a_{13}k_2 - k_1)g_{002},\\&e_3 = -a_{13}k_2(\lambda _{12} - 1)g_{011} + a_{32}(a_{11}k_1 + a_{13}k_2 - k_1)g_{002}.\\&b_1 = -\frac{(k_1 a_{11} + k_2 a_{13} - k_1\lambda _{12})e_1}{k_2 a_{13}(\lambda _{12} - 1)},\\&b_2 = \frac{-k_1 a_{32}(\lambda _{12} - 1)[(2a_{11}-2)g_{002} - a_{13}g_{101}] + e_2}{k_2a_{13}a_{32}(\lambda _{12} - 1)},\\&b_3 = -\frac{-k_1 a_{32}g_{002}(\lambda _{12} - 1) + e_3}{k_2a_{13}a_{32}(\lambda _{12} - 1)},\\ \end{aligned}$$

$$\begin{aligned}&b_4 = -\frac{(k_1 a_{11} + k_2 a_{13} - k_1\lambda _{12})[2a_{13}^2g_{200} - a_{13}(2a_{11} - \lambda _{12} - 1)g_{101} + 2(a_{11} - 1)(a_{11} - \lambda _{12})g_{002}]}{k_2 a_{13}(\lambda _{12} - 1)},\\&b_5 = \frac{-k_2 a_{13}(\lambda _{12} - 1)(a_{11} \!-\! \lambda _{12})g_{011}\! -\! a_{13}a_{32}(k_1 a_{11} \!+\! k_2 a_{13} \!-\! k_1\lambda _{12})g_{101} \!+\! 2a_{32}(a_{11}\! - \!\lambda _{12})(k_1 a_{11} \!+\! k_2 a_{13} \!-\! k_1\lambda _{12})g_{002}}{k_2a_{32}a_{13}(\lambda _{12} - 1)}.\\ \end{aligned}$$

Appendix B: $b_{ij}$ and $G_i(X_2,\delta _2,Y_2)$ in (3.12); $c_{i}$ in (3.16)

$$\begin{aligned}&b_{11} = 1 + \delta ^*_2(k_1A - l_1),~~ b_{13} = -\delta ^*_2 k_1B,\\&b_{31} = \delta ^*_2 k_2A,~~b_{33} = 1 - \delta ^*_2(k_2B + l_2),\\&G_1(X_2,\delta _2,Y_2) = {m}_{200}X_2^2 + {m}_{110}X_2\delta _2 + {m}_{101}X_2Y_2 \\&\qquad + {m}_{011}\delta _2Y_2 + {m}_{002}Y_2^2+ \mathcal {O}\left( \Vert (X_2,\delta _2,Y_2)\Vert ^3\right) ,\\&G_2(X_2,\delta _2,Y_2) = {n}_{200}X_2^2 + {n}_{110}X_2\delta _2 + {n}_{101}X_2Y_2 \\&\qquad + {n}_{011}\delta _2Y_2 + {n}_{002}Y_2^2+ \mathcal {O}\left( \Vert (X_2,\delta _2,Y_2)\Vert ^3\right) ,\\&m_{200} = \frac{k_1pq\left( -3q^2x^{*2} \!- \!3qx^{*2}y^* \!+\! pq^2 \!+\! 2pqy^* \!+\! py^{*2}\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ^*_2,\\&m_{110} = k_1A - l_1, \end{aligned}$$

$$\begin{aligned}&m_{101} = \frac{-2x^*k_1pq\left( -qx^{*2} + pq + py^*\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ^*_2,\\&m_{011} = -k_1B,~~ m_{002} = \frac{k_1qp^2x^{*2}}{\left( qx^{*2} + pq + py^*\right) ^3}\delta ^*_2,\\&n_{200} = \frac{k_2pq\left( -3q^2x^{*2} \!-\! 3qx^{*2}y^* \!+\! pq^2 \!+\! 2pqy^* \!+\! py^{*2}\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ^*_2,\\&n_{110} = k_2A,\\&n_{101} = \frac{2x^*k_2pq\left( -qx^{*2} + pq + py^*\right) }{\left( qx^{*2} + pq + py^*\right) ^3}\delta ^*_2,\\&n_{011} = -k_2B - l_2,~~ n_{002} = \frac{k_2qp^2x^{*2}}{\left( qx^{*2} + pq + py^*\right) ^3}\delta ^*_2. \end{aligned}$$

$$\begin{aligned}&c_1 = -\frac{\left( k_1 b_{11} + k_2 b_{13} + k_1\right) \left[ b_{13}^2n_{200} - b_{13}(b_{11} + 1)n_{101} + (b_{11} + 1)(b_{11}-\lambda _{22})n_{002}\right] }{b_{13}\left( \lambda _{22}^2 - 1\right) k_2},~~\\&c_2 = \frac{b_{13}^2n_{110}+(m_{110}-n_{011})b_{11}-\lambda _{22}(m_{110}-n_{011})b_{13} -m_{011}(b_{11}+1)(b_{11}-\lambda _{22} )}{b_{13}\left( \lambda _{22}^2 - 1\right) },\\&c_4 = -\frac{\left( k_1 b_{11} + k_2 b_{13} + k_1\right) \left[ 2b_{13}^2n_{200} - b_{13}(2b_{11} + 1-\lambda _{22})n_{101} + 2(b_{11} + 1)(b_{11}-\lambda _2)n_{002}\right] }{b_{13}\left( \lambda _{22}^2 - 1\right) k_2},~~\\&c_5 = - \frac{-b_{13}^2n_{110}\!-\!(m_{110}-n_{011})(b_{11}\!-\!\lambda _2)b_{13}\!-\!m_{011}(b_{11}\!-\!\lambda _{22})^2}{b_{13}\left( \lambda _{22}^2 - 1\right) },\\&c_3 = 0,~~ c_6 = 0.\\ \end{aligned}$$

Appendix C: $c_{ij}$ and $P_1(X_3,Y_3)$ in (3.18); E and all partial derivatives of $G_1$ and $G_2$ in (3.20)

$$\begin{aligned}&c_{11} = 1 + \delta (k_1A - l_1),~~ c_{12} = -\delta k_1B,~~ c_{21} = \delta k_2A,\\&c_{22} = 1-\delta (k_2B + l_2),\\&P_1(X_3,Y_3) = p_{20}X_3^2 + p_{11}X_3Y_3 + p_{02}Y_3^2 + \mathcal {O}(3),\\&p_{20} \!=\! \frac{k_1pq(-\!3q^2x^{*2} \!-\! 3qx^{*2}y^* \!+\! pq^2 \!+\! 2pqy^*\! +\! py^{*2})}{\left( qx^{*2} + pq + py^*\right) ^3}\delta ,\\ \end{aligned}$$

$$\begin{aligned}&p_{11} = -\frac{2k_1x^*pq(-qx^{*2} + pq + py^*)}{\left( qx^{*2} + pq + py^*\right) ^3}\delta ,~\\&p_{02} = \frac{k_1qp^2x^{*2}}{\left( qx^{*2} + pq + py^*\right) ^3}\delta .\\&E = \alpha ^2 - 3\alpha ^3 + 2\alpha ^4 - \beta ^2 + \alpha \beta ^2 - 2\beta ^4,\\&F = -\beta (\beta ^2 + 5\alpha ^2 - 2\alpha - 4\alpha ^3 - 4\alpha \beta ^2),\\ G_{1\widehat{X}_3\widehat{X}_3}&= \frac{2}{c_{12}}\left[ c_{12}^2f_{20}+c_{12}(\alpha -c_{11})f_{11}\right. \\&\left. +(\alpha -c_{11})^2f_{02}\right] ,\\ G_{1\widehat{Y}_3\widehat{Y}_3}&= \frac{2\beta ^2f_{02}}{c_{12}},\\ G_{1\widehat{X}_3\widehat{Y}_3}&= -\frac{\beta }{c_{12}}\left[ c_{12}f_{11} + 2(\alpha -c_{11})f_{02}\right] ,\\ G_{1\widehat{X}_3\widehat{X}_3\widehat{X}_3}&= \frac{6}{c_{12}}\left[ c_{12}^3f_{30}+c_{12}^2(\alpha -c_{11})f_{21} \right. \\&\qquad \left. + c_{12}(\alpha -c_{11})^2f_{12}+(\alpha -c_{11})^3f_{03}\right] ,\\ G_{1\widehat{X}_3\widehat{X}_3\widehat{Y}_3}&= -\frac{2\beta }{c_{12}}\left[ c_{12}^2f_{21}+2c_{12}(\alpha -c_{11})f_{12}\right. \\&\qquad \left. +3(\alpha -c_{11})^2f_{03}\right] ,\\&G_{1\widehat{X}_3\widehat{Y}_3\widehat{Y}_3}=\frac{2\beta ^2}{c_{12}}[c_{12}f_{12}+3(\alpha -c_{11})f_{03}],\\&G_{1\widehat{Y}_3\widehat{Y}_3\widehat{Y}_3}=-\frac{6\beta ^3f_{03}}{c_{12}}. \end{aligned}$$

Appendix D: $f_{ij}$ in (4.3); $\tilde{f}_{ij}$ in (4.5); $m_{ij}$ and $n_{ij}$ in (4.7); $U_2$, $V_2$, $W_1$, $R_1$, $W_2$, $R_2$, $V_3$ in (4.10)

$$\begin{aligned}&f_{20}= \frac{pq\left( -3q^2x^{*2}-3qx^{*2}y^*+pq^2+2pqy^*+py^{*2}\right) }{(qx^{*2}+pq+py^*)^3},\\&f_{11}= -\frac{2x^*pq\left( -qx^{*2}+pq+py^*\right) }{\left( qx^{*2}+pq+py^*\right) ^3},~~\\&f_{02}=\frac{qp^2x^{*2}}{\left( qx^{*2}+pq+py^*\right) ^3}, \end{aligned}$$

$$\begin{aligned}&f_{30}=-\frac{4xq^2p\left( -q^2x^{*2}-qx^{*2}y^*+pq^2+2pqy^*+py^{*2}\right) }{\left( qx^{*2}+pq+py^*\right) ^4},\\&f_{21}=-\frac{pq\left( 3q^2x^{*4}-8pq^2x^{*2}-8pqx^{*2}y^*+p^2q^2+2p^2qy^*+p^2y^{*2}\right) }{\left( qx^{*2}+pq+py^*\right) ^4}, \end{aligned}$$

$$\begin{aligned}&f_{12}=\frac{2p^2*x^*q\left( -2qx^{*2}+pq+py^*\right) }{\left( qx^{*2}+pq+py^*\right) ^4},\\&f_{03}=-\frac{p^3x^{*2}q}{\left( qx^{*2}+pq+py^*\right) ^4}.\\&\tilde{f}_{20} = d_{12}^2f_{20} + d_{11}d_{12}f_{11} + d_{11}^2f_{02}, \\&\tilde{f}_{11} = -2d_{12}^2\delta _4^*f_{20} - 2d_{12}\left( d_{11}\delta _4^* + 1\right) f_{11} \\&\qquad \quad - 2d_{11}\left( d_{11}\delta _4^* + 2\right) f_{02},\\&\tilde{ f}_{02} = d_{12}^2\delta _4^{*2}f_{20} + d_{12}\delta _4^*\left( d_{11}\delta _4^* + 2\right) f_{11}\\&\qquad \quad + \left( d_{11}\delta _4^* + 2\right) ^2f_{02},\\&\tilde{f}_{30} = d_{12}^3f_{30} + d_{11}d_{12}^2f_{21} + d_{11}^2d_{12}f_{12} + d_{11}^3f_{03},\\&\tilde{f}_{21} = -3d_{12}^3\delta _4^*f_{30} - \left( 3d_{11}\delta _4^* + 2\right) d_{12}^2f_{21}\\&\qquad \quad - d_{12}d_{11}\left( 3d_{11}\delta _4^* + 4\right) f_{12} - 3d_{11}^2\left( d_{11}\delta _4^* + 2\right) f_{03},~\\&\tilde{f}_{12} = 3d_{12}^3\delta _4^{*2}f_{30} + d_{12}^2\delta _4^*\left( 3d_{11}\delta _4^* + 4\right) f_{21}\\&\qquad \quad + d_{12}\left( d_{11}\delta _4^* + 2\right) \left( 3d_{11}\delta _4^* + 2\right) f_{12} \\&\qquad \quad + 3d_{11}\left( d_{11}\delta _4^* + 2\right) ^2f_{03},~~\\&\tilde{f}_{03} = -d_{12}^3\delta _4^{*3}f_{30} - d_{12}^2\delta _4^{*2}\left( d_{11}\delta _4^* + 2\right) f_{21} \\&\qquad \quad -\delta _4^*d_{12}\left( d_{11}\delta _4^* + 2\right) ^2f_{12} - \left( d_{11}\delta _4^* + 2\right) ^3f_{03}.\\&m_{20} = \left( P_1-\frac{P P_2}{Q} \right) \tilde{f}_{20},\\&m_{11} = \left( 2P_1-\frac{2P^2 P_2}{Q} \right) \tilde{f}_{20}+ \left( P_1 Q-P P_2 \right) \tilde{f}_{11},~~\\&m_{02} = \left( P_1 P^2-\frac{P^3 P_2}{Q} \right) \tilde{f}_{20}+\left( P_1 P Q-P^2 P_2 \right) \tilde{f}_{11}\\&\qquad \quad + \left( P_1 Q^2-P Q P_2 \right) \tilde{f}_{02},\\&m_{30} = \left( P_1-\frac{P P_2}{Q} \right) \tilde{f}_{30},~\\&m_{21} = \left( 3P_1-\frac{3P^2 P_2}{Q} \right) \tilde{f}_{30}+ \left( P_1 Q-P P_2 \right) \tilde{f}_{21},~~\\&m_{12} = \left( 3P_1 P^2-\frac{3P^3 P_2}{Q} \right) \tilde{f}_{30}\\&\qquad \quad +\left( 2P_1 P Q-2P^2 P_2 \right) \tilde{f}_{21}\\&\qquad \quad + \left( P_1 Q^2-P Q P_2 \right) \tilde{f}_{12},~\\&m_{03} = (P_1 P^3-\frac{P^4 P_2}{Q})\tilde{f}_{30}+(-P^3P_2+P^2 P_1 Q)\tilde{f}_{21}\\&\qquad \quad +(-P^2 P_2 Q+P P_1 Q^2)\tilde{f}_{12}\\&\quad \qquad +(P_1 Q^3 - P P_2 Q^2)\tilde{f}_{03},\\&n_{20} = \frac{ P_2}{Q}\tilde{f}_{20},~~ n_{11} = \frac{2P P_2}{Q}\tilde{f}_{20}+P_2 \tilde{f}_{11},\\&n_{02} =\frac{P^2 P_2}{Q}\tilde{f}_{20}+P P_2\tilde{f}_{11}+P_2 Q\tilde{f}_{02},\\&n_{30} = \frac{ P_2}{Q}\tilde{f}_{30},~~ n_{21} = \frac{3P P_2}{Q}\tilde{f}_{30}+P_2\tilde{f}_{21},\\&n_{12} = \frac{3P^2P_2}{Q} \tilde{f}_{30}+2P P_2\tilde{f}_{21}+P_2 Q\tilde{f}_{12},\\&n_{03} = \frac{P^3 P_2}{Q}\tilde{f}_{30}+P^2 P_2\tilde{f}_{21}+PQ P_2\tilde{f}_{12}+P_2 Q^2\tilde{f}_{03},\\&P = \delta _4[(d_{11}-d_{22})\delta _4^*+2],~~ Q = -2+\delta _4(d_{11}-d_{22}).\\&U_2 = -\frac{\delta _4^*x^*\left[ \left( k_1A-l_{1,4}^*\right) \delta _4^*+2\right] }{2k_1B},\\&V_2 = \frac{\left( k_1A - l_{1,4}^*\right) \delta _4^*x^*}{4k_1B},~\\&W_1 = -\frac{3q\delta _4^*p}{2\left[ \left( q + y^*\right) p + qx^{*2}\right] ^3d_{12}}\\&\qquad \quad \left\{ -2d_{11}d_{12}x^*\left( d_{11}\delta _4^*+\frac{4}{3}\right) \right. \\&\qquad \quad \left[ \left( \frac{1}{2}k_2x^*+k_1(q+y^*)\right) p - qk_1x^{*2}\right] \\&\quad \qquad +\left[ \left( q + y^*\right) \left( 2k_2x^*+k_1(q+y^*)\right) p \right. \\&\quad \qquad \left. - 3\left( \frac{2}{3}k_2 x^* + k_1\left( q+y^*\right) \right) q x^{*2}\right] \\&\quad \qquad \times \left( d_{11}\delta _4^* + \frac{2}{3}\right) d_{12}^2+ pk_1 x^{*2}d_{11}^2 \left( d_{11} \delta _4^*+2\right) \\&\qquad \quad \left. \delta _4^*k_2d_{12}^3\left[ \left( q + y^*\right) p - 3qx^{*2}\right] \times \left( q+y^*\right) \right\} ,\\&R_1 = -\frac{3q\delta _4^*p}{\left[ \left( q + y^*\right) p + qx^{*2}\right] ^3d_{12}}\\&\quad \times \left\{ \delta _4^*d_{12}^2\left( d_{11}\delta _4^* + \frac{4}{3}\right) \left[ -2qx^{*3}k_2 -3qk_1\left( q+y^*\right) x^{*2} \right. \right. \\&\quad \left. + 2pk_2\left( q +y^*\right) x^*+ pk_1\left( q+y^*\right) ^2\right] \\&\quad -2x^*d_{12}\left( d_{11}\delta _4^* + 2\right) \left( d_{11}\delta _4^*+\frac{2}{3}\right) \\&\quad \left( -qk_1x^{*2} + \frac{1}{2}pk_2x^* + pk_1\left( q+y^*\right) \right) \\&\quad -\delta _4^{*2}k_2d_{12}^3\left[ \left( q+y^*\right) p - 3qx^{*2}\right] \left( q+y^*\right) {\left( y^*\right) ^2}\\&\quad \left. + p k_1 x^{*2} d_{11}\left( d_{11} \delta _4^*+2\right) ^2 \right\} , \end{aligned}$$

$$\begin{aligned}&W_2 = -\frac{q\delta _4^*p}{2\left[ \left( q + y^*\right) p + qx^{*2}\right] ^3d_{12}}\\&\quad \times \left\{ -\delta _4^{*}k_2d_{12}^3\left[ \left( q + y^*\right) p - 3qx^{*2}\right] \left( q+y^*\right) \right. \\&\quad + pk_1 x^{*2} d_{11}^2\left( d_{11}\delta _4^*+ 2\right) + d_{12}^2 \left[ \left( q + y^*\right) (k_1\left( d_{11} \delta _4^* - 2\right) q\right. \\&\quad \left. +\left( 2d_{11}\delta _4^*k_2 + 4k_2\right) x^* + k_1y^*\left( d_{11}\delta _4^*-2\right) \right) \\&\quad \times p - 3qx^{*2}\left( k_1(d_{11} \delta _4^*-2)q +\frac{2}{3}k_2\left( d_{11} \delta _4^*+2\right) x^*\right. \\&\quad \left. \left. +k_1y^*\left( d_{11}\delta _4^*-2\right) \right) \right] \\&\quad -2d_{11}x^*d_{12} \left[ \left( qk_1d_{11} \delta _4^* + \left( \frac{1}{2}d_{11} \delta _4^*k_2 + 2k_2\right) x^* \right. \right. \\&\quad \left. \left. \left. +k_1y^*d_{11} \delta _4^*\right) p - qk_1d_{11} \delta _4^*x^{*2}\right] \right\} ,\\&R_2 = \frac{q\delta _4^*p}{\left[ \left( q + y^*\right) p + qx^{*2}\right] ^3d_{12}}\\&\quad \times \left\{ -\delta _4^{*2}k_2d_{12}^3\left[ \left( q + y^*\right) p - 3qx^{*2}\right] ^3\left( q+y^*\right) \right. \\&\quad + pk_1 x^{*2}d_{11}\left( d_{11} \delta _4^*+2\right) ^2-2x^*d_{12}\left( d_{11}\delta _4^*+2\right) \\&\quad \times \left[ d_{11}\delta _4^*\left( \left( \frac{1}{2}k_2x^*+k_1\left( q+y^*\right) \right) p-qk_1x^{*2}\right) \right. \\&\quad \left. +\left( -k_2x^* +2k_1\left( q+y^*\right) \right) p-2qk_1x^{*2} {\frac{1}{2}}\right] \\&\quad +\delta _4^*d_{12}^2\left[ \left( d_{11}\delta _4^*\left( \left( q+y^*\right) \left( 2k_2x^* +k_1\left( q+y^*\right) \right) p \right. \right. \right. \\&\quad \left. -3qx^{*2}\left( \frac{2}{3}k_2x^*+k_1\left( q+y^*\right) \right) \right) \\&\quad \left. \left. +4k_1\left( \left( q+y^*\right) p-3qx^{*2}\right) \left( q+y^*\right) \right] {\left( y^*\right) ^2} {(\frac{1}{2})^2\left[ \delta ^{*2}\right] ^2}\right\} ,\\&V_{3} = k_1A-k_2B-l_{1,3}^*-l_2.\\ \end{aligned}$$

Appendix E: $\widehat{f}_{ij}$ in (4.14); $\xi _{ij}$ and $\zeta _{ij}$ in (4.16); $C(\theta )$ and $D(\theta )$ in (4.17); $C_1(\theta )$ and $D_1(\theta )$ in (4.21)

$$\begin{aligned} \widehat{f}_{20}&= \left( d_{11}^2f_{02}+d_{11}d_{12}f_{11}+d_{12}^2f_{20}\right) \delta ^2\\&\quad + \left( 4d_{11}f_{02}+2d_{12}f_{11}\right) \delta + 4f_{02},\\ \widehat{f}_{11}&= \left( -2d_{11}f_{02}-d_{12}f_{11}\right) \delta - 4f_{02},\\ \widehat{f}_{30}&= \left( -d_{11}^3f_{03}-d_{11}^2d_{12}f_{12}-d_{11}d_{12}^2f_{21}-d_{12}^3f_{30}\right) \delta ^3\\&\quad + \left( -6d_{11}^2f_{03}-4d_{11}d_{12}f_{12}-2d_{12}^2f_{21}\right) \delta ^2\\&\quad + \left( -12d_{11}f_{03}-4d_{12}f_{12}\right) \delta - 8f_{03},\\ \widehat{f}_{21}&= \left( 3d_{11}^2f_{03}+2d_{11}d_{12}f_{12}+d_{12}^2f_{21}\right) \delta ^2\\&\quad + \left( 12d_{11}f_{03}+4d_{12}f_{12}\right) \delta + 12f_{03},\\ \widehat{f}_{12}&= \left( -3d_{11}f_{03}-d_{12}f_{12}\right) \delta - 6f_{03},\\&\quad \widehat{f}_{03} = f_{03},~~ \widehat{f}_{02}=f_{02}.\\ \xi _{20}&=\frac{1}{4}\widehat{f}_{20}(2P_1+P_2),~~ \xi _{11}=\frac{1}{4}(2P_1+P_2)\widehat{f}_{11}\\&\quad +\frac{1}{2}\widehat{f}_{20}(P_1+P_2),~~\\ \xi _{02}&=\frac{1}{8}\left( 2\widehat{f}_{02}+2\widehat{f}_{11}+ \widehat{f}_{20}\right) P_2\\&\quad +\frac{1}{2}\left( \frac{1}{2}\widehat{f}_{11} +\widehat{f}_{02}\right) P_1,\\ \xi _{30}&=\left( \frac{1}{4}\widehat{f}_{02}\widehat{f}_{11}+\frac{1}{4} \widehat{f}_{02}\widehat{f}_{20} +\frac{1}{12}\widehat{f}_{11}^2\right. \\&\quad \left. +\frac{1}{6}\widehat{f}_{11}\widehat{f}_{20}+\frac{1}{24}\widehat{f}_{20}^2\right) P_2^2\\&\quad +\left( \frac{1}{2}\widehat{f}_{11}\widehat{f}_{20}-\frac{1}{6}\widehat{f}_{20}^2\right) P_1^2 \\&\quad +\left( \frac{1}{2}\widehat{f}_{02}\widehat{f}_{20}+\frac{1}{4}\widehat{f}_{11}^2+\frac{1}{3}\widehat{f}_{11}\widehat{f}_{20} +\frac{1}{12}\widehat{f}_{20}^2\right) P_1P_2\\&\quad +\frac{1}{6}P_2\widehat{f}_{12}+\left( \frac{1}{3}\widehat{f}_{21}-\frac{1}{2}\widehat{f}_{30}\right) P_1,\\ \xi _{21}&=\left( \widehat{f}_{02}\widehat{f}_{11}+\frac{7}{4}\widehat{f}_{02}\widehat{f}_{20}+\frac{7}{8}\widehat{f}_{11}^2+\frac{9}{8}\widehat{f}_{11}\widehat{f}_{20}\right. \\&\quad \left. +\frac{1}{8}\widehat{f}_{20}^2\right) P_1P_2 +\left( \frac{1}{2}\widehat{f}_{03}+\frac{1}{2}\widehat{f}_{12}\right) P_2 \\&\quad +\left( \frac{1}{2}\widehat{f}_{12}+\frac{1}{2}\widehat{f}_{21}-\widehat{f}_{30}\right) P_1\\&\quad +\left( \frac{1}{2}\widehat{f}_{02}\widehat{f}_{20} +\frac{1}{4}\widehat{f}_{11}^2+\frac{5}{4} \widehat{f}_{11}\widehat{f}_{20}\frac{1}{2}\widehat{f}_{20}^2\right) P_1^2 \\&\quad +\left( \frac{1}{2}\widehat{f}_{02}^2+\frac{9}{8}\widehat{f}_{02}\widehat{f}_{11} +\frac{3}{4}\widehat{f}_{02}\widehat{f}_{20}+\frac{3}{8}\widehat{f}_{11}^2\right. \\&\quad \left. +\frac{9}{16}\widehat{f}_{11}\widehat{f}_{20}+\frac{1}{8}\widehat{f}_{20}^2\right) P_2^2,\\ \xi _{12}&=\left( \frac{11}{12}\widehat{f}_{11}\widehat{f}_{20}+\frac{5}{4}\widehat{f}_{02}\widehat{f}_{20}+\frac{7}{4}\widehat{f}_{02}\widehat{f}_{11}\right. \\&\quad \left. +\frac{1}{24}\widehat{f}_{20}^2+\widehat{f}_{02}^2 +\frac{7}{8}\widehat{f}_{11}^2\right) P_1P_2\\&\quad +\left( \frac{1}{2}\widehat{f}_{03}+\frac{1}{3}\widehat{f}_{12}\right) P_2 \\&\quad +\left( \frac{1}{2}\widehat{f}_{12}+\frac{1}{6}\widehat{f}_{21}-\frac{1}{2}\widehat{f}_{30}+\widehat{f}_{03}\right) P_1\\&\quad +\left( -\frac{1}{3}\widehat{f}_{20}^2+\frac{1}{2}\widehat{f}_{11}^2 +\frac{1}{2}\widehat{f}_{02}\widehat{f}_{20}+\frac{1}{2}\widehat{f}_{02}\widehat{f}_{11} +\frac{3}{4}\widehat{f}_{11}\widehat{f}_{20}\right) P_1^2 \\&\quad +\left( \frac{1}{12}\widehat{f}_{20}^2+\frac{7}{24}\widehat{f}_{11}^2+\frac{1}{2}\widehat{f}_{02}^2+\frac{19}{48}\widehat{f}_{11}\widehat{f}_{20}\right. \\&\quad \left. +\frac{7}{8}\widehat{f}_{02}\widehat{f}_{11} +\frac{1}{2}\widehat{f}_{02}\widehat{f}_{20}\right) P_2^2,\\ \zeta _{20}&=\frac{1}{2}P_2\widehat{f}_{20},~~ \zeta _{11}=\frac{1}{2}P_2\left( \widehat{f}_{11}+\widehat{f}_{20}\right) ,\\ \zeta _{02}&= \frac{1}{4}P_2\left( 2\widehat{f}_{02} + \widehat{f}_{11}\right) ,~~\\ \zeta _{30}&= -\left[ \left( P_1+\frac{1}{2}P_2\right) \widehat{f}_{20}^2+\frac{1}{2}\widehat{f}_{11}P_2\widehat{f}_{20}+\widehat{f}_{30}\right] P_1,\\&\quad \zeta _{03}(\theta )=\xi _{03}(\theta )=0,\\ \end{aligned}$$

$$\begin{aligned} \zeta _{21}&=\frac{1}{2}\left[ \widehat{f}_{20}^2+\left( 6\widehat{f}_{02}+4\widehat{f}_{11}\right) \widehat{f}_{20}+6\widehat{f}_{02}\widehat{f}_{11}+2\widehat{f}_{11}^2\right] P_2^2\\&\quad +\frac{1}{8}\left[ -6P_1\widehat{f}_{20}^2+\left( 4\widehat{f}_{02}-\frac{1}{2}\widehat{f}_{11}\right) P_1\widehat{f}_{20} \right. \\&\quad \left. +2P_1\widehat{f}_{11}^2+4\widehat{f}_{12}\right] P_2 -\frac{3}{2}P_1\left( P_1\widehat{f}_{20}^2+\widehat{f}_{30}\right) ,\\ \zeta _{12}&=\left( \frac{1}{2}\widehat{f}_{11}+\frac{1}{4}\widehat{f}_{20}+\widehat{f}_{02}\right) \left( \widehat{f}_{11}+\frac{1}{2}\widehat{f}_{20}+\widehat{f}_{02}\right) P_2^2\\&\quad +\frac{1}{8}\left[ 4P_1\widehat{f}_{11}^2+\left( 4P_1\widehat{f}_{02}+4\widehat{f}_{20}\right) \widehat{f}_{11} \right. \\&\quad \left. +4P_1\widehat{f}_{02}\widehat{f}_{20}+8\widehat{f}_{03}+4\widehat{f}_{12}\right] P_2-\left( \widehat{f}_{20}\widehat{f}_{20}+\frac{1}{2}\widehat{f}_{30}\right) P_1.\\ C(\theta )&=\left[ \left( \frac{1}{4}\widehat{f}_{02}\widehat{f}_{11} + \frac{1}{4}\widehat{f}_{20}\widehat{f}_{02} + \frac{1}{12}\widehat{f}_{11}^2 + \frac{1}{6}\widehat{f}_{20}\widehat{f}_{11}\right. \right. \\&\quad \left. +\frac{1}{24}\widehat{f}_{20}^2\right) P_2^2 +\left( \frac{1}{3}\widehat{f}_{21} - \frac{1}{2}\widehat{f}_{30}\right) P_1 \\&\quad + \left( \frac{1}{2}\widehat{f}_{20}\widehat{f}_{11} -\frac{1}{6}\widehat{f}_{20}^2\right) P_1^2+\frac{1}{6}P_2\widehat{f}_{12}\\&\quad + \left( \frac{1}{2}\widehat{f}_{20}\widehat{f}_{02} + \frac{1}{4}\widehat{f}_{11}^2 +\frac{1}{3}\widehat{f}_{20}\widehat{f}_{11}+\frac{1}{12}\widehat{f}_{20}^2\right) \\&\quad \left. \times P_1P_2 {\frac{1}{4}\widehat{f}_{02}\widehat{f}_{11}} \right] \theta _1 \\&\quad + \left[ \left( -\frac{1}{2}\widehat{f}_{20}\widehat{f}_{11} -\frac{1}{2}\widehat{f}_{20}^2\right) P_1P_2-P_1\widehat{f}_{30}-P_1^2\widehat{f}_{20}^2\right] \theta _2\\&\quad +P_1P_2\widehat{f}_{20}^2 +P_2\widehat{f}_{30}+\left( \frac{1}{2}\widehat{f}_{20}\widehat{f}_{11} + \frac{1}{2}\widehat{f}_{20}^2\right) P_2^2 ,\\ \end{aligned}$$

$$\begin{aligned} D(\theta )&=\left[ \left( \frac{1}{2}\widehat{f}_{03} + \frac{1}{2}\widehat{f}_{12}\right) P_2 + \left( \widehat{f}_{02}\widehat{f}_{11} + \frac{7}{4}\widehat{f}_{20}\widehat{f}_{02}\right. \right. \\&\quad \left. + \frac{7}{8}\widehat{f}_{11}^2 +\frac{9}{8}\widehat{f}_{20}\widehat{f}_{11} + \frac{1}{8}\widehat{f}_{20}^2\right) P_1P_2 \\&\quad +\left( \frac{1}{2}\widehat{f}_{12} +\frac{1}{2}\widehat{f}_{21}-\widehat{f}_{30}\right) P_1 +\left( \frac{1}{2}\widehat{f}_{02}^2+\frac{9}{8}\widehat{f}_{02}\widehat{f}_{11}\right. \\&\quad \left. +\frac{3}{4}\widehat{f}_{20}\widehat{f}_{02} +\frac{3}{8}\widehat{f}_{11}^2 +\frac{9}{16}\widehat{f}_{20}\widehat{f}_{11}+\frac{1}{8}\widehat{f}_{20}^2\right) P_2^2\\&\quad \left. +\left( \frac{1}{2}\widehat{f}_{20}\widehat{f}_{02}+\frac{1}{4}\widehat{f}_{11}^2 +\frac{5}{4}\widehat{f}_{20}\widehat{f}_{11} -\frac{1}{2}\widehat{f}_{20}^2\right) P_1^2\right] \theta _1\\&\quad +\left[ \frac{1}{2}P_2\widehat{f}_{12} +\left( \frac{1}{2}\widehat{f}_{20}\widehat{f}_{02}+\frac{1}{4}\widehat{f}_{11}^2 -\frac{1}{4}\widehat{f}_{20}\widehat{f}_{11} -\frac{3}{4}\widehat{f}_{20}^2\right) P_1P_2\right. \\&\quad +\left( \frac{3}{4}\widehat{f}_{02}\widehat{f}_{11} {+}\frac{3}{4}\widehat{f}_{20}\widehat{f}_{02} +\frac{1}{4}\widehat{f}_{11}^2+\frac{1}{2}\widehat{f}_{20}\widehat{f}_{11}{+}\frac{1}{8}\widehat{f}_{20}^2\right) P_2^2\\&\left. \quad -\frac{3}{2}P_1\widehat{f}_{30} -\frac{3}{2}P_1^2\widehat{f}_{20}^2\right] \theta _2\\&\quad +\left( 2\widehat{f}_{20}\widehat{f}_{11}+\frac{3}{2}\widehat{f}_{20}^2\right) P_2P_1\\&\quad +\left[ \left( \frac{1}{2}\widehat{f}_{11}+\frac{1}{2}\widehat{f}_{20}\right) \widehat{f}_{11}+\widehat{f}_{20}\widehat{f}_{02} +\frac{1}{4}\widehat{f}_{20}\widehat{f}_{11}\right] P_2^2 \\&\quad +3P_1^2\widehat{f}_{20}^2+3P_1\widehat{f}_{30}.\\ C_1(\theta )&=4\left( \left( P_1+\frac{1}{2}P_2\right) \widehat{f}_{20}^2+\frac{1}{2}P_2\widehat{f}_{11}\widehat{f}_{20}+\widehat{f}_{30}\right) P_2\\&\quad +\left[ \left( -\frac{5}{3}\widehat{f}_{11}^2-\frac{4}{3}\widehat{f}_{20}^2{+}\widehat{f}_{02}\widehat{f}_{11} {-}3\widehat{f}_{02}\widehat{f}_{20}{-}\frac{31}{6}\widehat{f}_{11}\widehat{f}_{20}\right) P_2^2 \right. \\&\quad +\left( -\frac{38}{3}\widehat{f}_{20}^2+2\widehat{f}_{11}\widehat{f}_{20}\right) P_1^2+\left( \frac{4}{3}\widehat{f}_{21}-14\widehat{f}_{30}\right) P_1 \\&\quad +\left( \frac{2}{3}\widehat{f}_{12}-4\widehat{f}_{21} -\frac{5}{3}\widehat{f}_{30}\right) P_2\\&\quad \left. +\left( 2\widehat{f}_{02}\widehat{f}_{20}-\frac{32}{3}\widehat{f}_{11}\widehat{f}_{20} +\widehat{f}_{11}^2 -\frac{34}{3}\widehat{f}_{20}^2\right) P_1P_2\right] \theta _1 \\&\quad +\left[ -4\left( \left( P_1+\frac{1}{2}P_2\right) \times \widehat{f}_{20}^2+\frac{1}{2}P_2\widehat{f}_{11}\widehat{f}_{20} +\widehat{f}_{30}\right) \right. \\&\quad \times \left. \left( P_1-\frac{1}{2}P_2\right) \right] \theta _2,\\ D_1(\theta )&=\left( -\frac{11}{2}\widehat{f}_{11}\widehat{f}_{20}-2\widehat{f}_{02}\widehat{f}_{20}-\widehat{f}_{11}^2-5\widehat{f}_{20}^2\right) P_2^2\\&\quad -6P_1^2\widehat{f}_{20}^2 +\left( -2\widehat{f}_{21}-6\widehat{f}_{30}\right) P_2 \\&\quad -6P_1\widehat{f}_{30} +\left( -6\widehat{f}_{11}\widehat{f}_{20}-11\widehat{f}_{20}^2\right) P_1P_2\\&\quad +\left[ \left( -\frac{15}{4}\widehat{f}_{02}\widehat{f}_{11}-\frac{4}{3}\widehat{f}_{02}\widehat{f}_{20} +\frac{5}{6}\widehat{f}_{11}\widehat{f}_{20} \right. \right. \\&\quad \left. -\widehat{f}_{02}^2-\frac{5}{12}\widehat{f}_{11}^2+\frac{49}{24}\widehat{f}_{20}^2\right) P_2^2\\&\quad + \left( -\widehat{f}_{02}\widehat{f}_{20}-\frac{11}{2}\widehat{f}_{11}\widehat{f}_{20} -\frac{1}{2}\widehat{f}_{11}^2 + 7\widehat{f}_{20}^2\right) P_1^2 \\&\quad +\left( -\widehat{f}_{12}-3\widehat{f}_{21}+10\widehat{f}_{30}\right) P_1\\&\quad +\left( \frac{5}{3}\widehat{f}_{21}+\frac{7}{4}\widehat{f}_{30} -\widehat{f}_{03}-2\widehat{f}_{12}\right) P_2\\&\quad +\left( \frac{3}{4}\widehat{f}_{11}\widehat{f}_{20}-2\widehat{f}_{02}\widehat{f}_{11}\right. \\&\quad \left. \left. -\frac{13}{2}\widehat{f}_{02}\widehat{f}_{20} -\frac{13}{4}\widehat{f}_{11}^2 +\frac{31}{6}\widehat{f}_{20}^2\right) P_1P_2\right] \theta _1\\&+\left[ \left( -\frac{3}{2}\widehat{f}_{02}\widehat{f}_{11} -\frac{1}{2}\widehat{f}_{02}\widehat{f}_{20} +\frac{1}{2}\widehat{f}_{11}\widehat{f}_{20}+\widehat{f}_{20}^2\right) P_2^2 \right. \\&\quad +12 P_1^2 \widehat{f}_{20}^2+12 P_1 \widehat{f}_{30} +\left( \widehat{f}_{21}+\frac{1}{2}\widehat{f}_{30}-\widehat{f}_{12}\right) P_2 \\&\quad \left. +\left( -\widehat{f}_{02} \widehat{f}_{20}+\frac{13}{2}\widehat{f}_{11} \widehat{f}_{20}-\frac{1}{2} \widehat{f}_{11}^2+\frac{15}{2} \widehat{f}_{20}^2\right) P_1 P_2\right] \theta _2.\\ \end{aligned}$$