Global analysis of the dynamics of a mathematical model to intermittent HIV treatment

Abstract

The aim of this paper is to study the qualitative dynamics of a piecewise smooth system modeling the intermittent treatment of the human immunodeficiency virus. Typical singularities and closed orbits are observable, and we quantitatively explore the dynamics around those singularities and closed orbits. Moreover, we conclude that this protocol always will be successful since the trajectory passing through any initial condition converges to one of these distinguished orbits. Our formal mathematical results corroborate the real-world observation, where the virus is not eliminated, but the number of infected cells is controlled around a specific value.

Appendix A. Computing the half-return maps

We consider a trajectory of system (30) initiated at the point ${\mathbf {x}}_0=(x_0,y_0,0)\in \Sigma ^{c,+}$ such that for $t=t_1>0$ this trajectory transversally returns to $\Sigma$ at the point ${\mathbf {x}}_1=(x_1,y_1,0)\in \Sigma ^{c,-}$. And its continuation from the point ${\mathbf {x}}_1$ such that for $t=t_2>0$ this trajectory transversally returns to $\Sigma$ at the point ${\mathbf {x}}_2=(x_2,y_2,0)\in \Sigma ^{c,+}$. See illustration in Fig. 12. Then, we define the half-return maps $(x_1,y_1)=P_+(x_0,y_0)$ and $(x_2,y_2)=P_-(x_1,y_1)$. In order to find the explicit expressions for the coefficients of each half-return map, we must determine the solution of system (30) for ${\widetilde{z}}=z- C_T\ge 0$ and for ${\widetilde{z}}=z-C_T\le 0$. Such solution for ${\widetilde{z}}\ge 0$ can be obtained from the variation in constants formula

Fig. 12

First return application in $\Sigma ^{c,+}$

$$\begin{aligned} \widetilde{{\mathbf {x}}}(t)= & {} e^{A^+t}\widetilde{{\mathbf {x}}}_0+\int _0^{t} e^{A^+(t-s)}{\mathbf {v}}^+ds\nonumber \\= & {} e^{A^+t}\widetilde{{\mathbf {x}}}_0+\left( \int _0^{t} e^{A^+s}ds\right) {\mathbf {v}}^+, \end{aligned}$$

(A.1)

where

$$\begin{aligned} A^{+}= & {} \begin{bmatrix} -\alpha - {\bar{y}}_p &{}-{\bar{x}}_t&{}0\\ -a &{} -c &{}a\\ \delta -\alpha &{}0&{}-\delta \\ \end{bmatrix},\;\;\\ \widetilde{{\mathbf {x}}}_0= & {} \begin{bmatrix} {\widetilde{x}}_0\\ {\widetilde{y}}_0\\ 0 \end{bmatrix},\;\;{\mathbf {v}}^{+}=\begin{bmatrix} {\bar{x}}_t({\bar{y}}_q^+ - {\bar{y}}_p)\\ \frac{a{\bar{x}}_t{\bar{y}}_q^+}{\delta }-c{\bar{y}}_p\\ 0 \end{bmatrix}\;\;\text {and}\;\;\\ \widetilde{{\mathbf {x}}}(t)= & {} \begin{bmatrix} {\widetilde{x}}(t)\\ {\widetilde{y}}(t)\\ {\widetilde{z}}(t) \end{bmatrix}=\begin{bmatrix} x(t)-{\bar{x}}_t\\ y(t)-{\bar{y}}_p\\ z(t)-C_T \end{bmatrix}, \end{aligned}$$

being ${\widetilde{x}}_0\ge 0$, $A^+=D{\mathbf {F}}_{+}({\bar{x}}_t,{\bar{y}}_p,C_T)$, ${\mathbf {v}}^{+}={\mathbf {F}}_{+}({\bar{x}}_t,{\bar{y}}_p,C_T)$. For simplicity, we take $C_T=\frac{{\bar{x}}_t{\bar{y}}_q^+}{\delta } + {\bar{x}}_t$ and $s={\bar{x}}_t\left( \alpha + {\bar{y}}_q^+\right)$. It will suffice to have the Taylor’s series approximation until the sixth-order terms for (A.1), namely

$$\begin{aligned} \widetilde{{\mathbf {x}}}(t)= & {} \widetilde{{\mathbf {x}}}_0+\left( t I+\dfrac{t^2}{2}A^++\dfrac{t^3}{6}(A^+)^2+\dfrac{t^4}{24}(A^+)^3\right. \nonumber \\&\left. + \dfrac{t^5}{5!}(A^+)^4+\dfrac{t^6}{6!}(A^+)^5\right) M+O(t^7), \end{aligned}$$

(A.2)

where $M=A^+\widetilde{{\mathbf {x}}}_0+{\mathbf {v}}^+$ and I is the identity matrix of order 3.

Remark 4

Note that solution (A.2) refers to system (30) linearized at pseudo-equilibrium point ${\mathbf {p}}$ [given in (35)] with ${\mathbf {p}}$ moved to the origin of a new state space $\mathcal {{\widetilde{M}}}=\{({\widetilde{x}},{\widetilde{y}}, {\widetilde{z}})\in {\mathbb {R}}^3: {\widetilde{x}}^2+{\widetilde{y}}^2+{\widetilde{z}}^2<\varepsilon ^2 \}$ for $\varepsilon ^2$ arbitrarily small. With this change the set $L_I$ of invisible two–folds is moved to $(0,{\widetilde{y}},0)$ and the crossing regions $\Sigma ^{c,+}$ and $\Sigma ^{c,-}$ are redefined for all $({\widetilde{x}},{\widetilde{y}},0)$ with ${\widetilde{x}}>0$ and ${\widetilde{x}}<0$, respectively.

From the third component of (A.2), we can determine an expression for the half-return time $t_+=t_+({\widetilde{x}}_0,{\widetilde{y}}_0)$, which depends on $({\widetilde{x}}_0,{\widetilde{y}}_0)$ and such that ${\widetilde{z}}(t_+)=0$. The polynomial approximation for the time $t_+$ is given by

$$\begin{aligned} t_+({\widetilde{x}}_0,{\widetilde{y}}_0)={\widetilde{x}}_0\cdot \Phi ({\widetilde{x}}_0,{\widetilde{y}}_0), \end{aligned}$$

(A.3)

where $\Phi (0,0)=\frac{2}{{\bar{x}}_t({\bar{y}}_p-{\bar{y}}_q^+)}>0$. For our purpose, we must compute all the coefficients of the polynomial $\Phi ({\widetilde{x}}_0,{\widetilde{y}}_0)$ up to the fourth order terms. Some of these coefficients are very large expressions, and therefore, we do not make them explicit.

Now, substituting (A.3) in the first and the second component of (A.2), the half-return map $({\widetilde{x}}_1,{\widetilde{y}}_1)=P_+({\widetilde{x}}_0,{\widetilde{y}}_0)$ satisfies

$$\begin{aligned} {\widetilde{x}}_1&=-{\widetilde{x}}_0+p_{20}^+{\widetilde{x}}_0^2 +p_{30}^+{\widetilde{x}}_0^3+p_{21}^+{\widetilde{x}}_0^2{\widetilde{y}}_0+ p_{40}^+{\widetilde{x}}_0^4\nonumber \\&\quad +p_{31}^+{\widetilde{x}}_0^3{\widetilde{y}}_0 +p_{22}^+{\widetilde{x}}_0^2{\widetilde{y}}_0^2 +{\widetilde{x}}_0^2\cdot O(3), \end{aligned}$$

(A.4)

$$\begin{aligned} {\widetilde{y}}_1&=v_{10}^+{\widetilde{x}}_0+{\widetilde{y}}_0+v_{11}^+ {\widetilde{x}}_0{\widetilde{y}}_0+v_{20}^+{\widetilde{x}}_0^2+v_{30}^+ {\widetilde{x}}_0^3\nonumber \\&\quad +v_{21}^+{\widetilde{x}}_0^2{\widetilde{y}}_0+v_{12}^+ {\widetilde{x}}_0{\widetilde{y}}_0^2+{\widetilde{x}}_0\cdot O(3), \end{aligned}$$

(A.5)

with coefficients of the quadratic terms given by

$$\begin{aligned} p_{20}^+= & {} \dfrac{\delta \left[ c{\bar{y}}_p+({\bar{y}}_p-{\bar{y}}_q^+) (\alpha +\delta +{\bar{y}}_p)\right] -a{\bar{x}}_t{\bar{y}}_q^+}{3/2{\bar{x}}_t \delta ({\bar{y}}_q^+-{\bar{y}}_p)^2} ,\nonumber \\ v_{10}^+= & {} \dfrac{2(a{\bar{x}}_t{\bar{y}}_q^+-c\delta {\bar{y}}_p)}{{\bar{x}}_t\delta ({\bar{y}}_p-{\bar{y}}_q^+)},\nonumber \\ v_{11}^+= & {} \dfrac{2{\bar{y}}_q^+(c\delta -a{\bar{x}}_t)}{{\bar{x}}_t \delta ({\bar{y}}_p-{\bar{y}}_q^+)^2},\nonumber \\ v_{20}^+= & {} \dfrac{2(a{\bar{x}}_t{\bar{y}}_q^+-c\delta {\bar{y}}_p) (2a{\bar{x}}_t{\bar{y}}_q^++\delta [c({\bar{y}}_p-3{\bar{y}}_q^+)+ ({\bar{y}}_p-{\bar{y}}_q^+)(\alpha +\delta +{\bar{y}}_p)])}{3{\bar{x}}_t^2 \delta ^2({\bar{y}}_q^+-{\bar{y}}_p)^3}, \end{aligned}$$

(A.6)

coefficients of the cubic terms given by

$$\begin{aligned} p_{21}^+=&\dfrac{\delta \left[ c({\bar{y}}_p+{\bar{y}}_q^+)+({\bar{y}}_p-{\bar{y}}_q^+) (\alpha +\delta +{\bar{y}}_p)\right] -2a{\bar{x}}_t{\bar{y}}_q^+}{3/2{\bar{x}}_t\delta ({\bar{y}}_q^+-{\bar{y}}_p)^3} ,\nonumber \\ p_{30}^+=&\frac{2}{9\delta ^2{\bar{x}}_t^2({\bar{y}}_q^+-{\bar{y}}_p)^4} \left( 4a^2{\bar{x}}_t^2({\bar{y}}_q^+)^2\right. \nonumber \\&\left. -a\delta {\bar{x}}_t{\bar{y}}_q^+ \left[ c(3{\bar{y}}_q^++5{\bar{y}}_p)+({\bar{y}}_q^+-{\bar{y}}_p)(\alpha +\delta +{\bar{y}}_p)\right] +\right. \nonumber \\&\left. -\delta ^2\left[ c^2(-{\bar{y}}_p)(3{\bar{y}}_q^++{\bar{y}}_p) +\alpha ({\bar{y}}_q^+-{\bar{y}}_p)\right. \right. \nonumber \\&\left. \left. (4({\bar{y}}_q^+-{\bar{y}}_p)(\delta +{\bar{y}}_p)\right. \right. \nonumber \\&\left. \left. -c{\bar{y}}_p)-c{\bar{y}}_p({\bar{y}}_q^+-{\bar{y}}_p)(\delta +{\bar{y}}_p)+\right. \right. \nonumber \\&\left. \left. +2\alpha ^2({\bar{y}}_q^+-{\bar{y}}_p)^2+2({\bar{y}}_q^ +-{\bar{y}}_p)^2(\delta +{\bar{y}}_p)^2\right] \right) ,\nonumber \\ v_{30}^+=&\frac{-2}{9 \delta ^3 {\bar{x}}_t^3 ({\bar{y}}_q^+-{\bar{y}}_p)^5} \left( 8 a^3 {\bar{x}}_t^3 ({\bar{y}}_q^+)^3\right. \nonumber \\&\left. -a^2 \delta {\bar{x}}_t^2 {\bar{y}}_q^+ (3 c {\bar{y}}_q^+ (5 {\bar{y}}_q^++3 {\bar{y}}_p)\right. \nonumber \\&\left. + ({\bar{y}}_q^+-{\bar{y}}_p) (5 {\bar{y}}_q^+ (\alpha +{\bar{y}}_p)+\right. \nonumber \\&\left. +\delta (2 {\bar{y}}_q^++3 {\bar{y}}_p)))+a \delta ^2 {\bar{x}}_t \left( 6 c^2 ({\bar{y}}_q^+)^2 ({\bar{y}}_q^+\right. \right. \nonumber \\&\left. \left. +3 {\bar{y}}_p)+c {\bar{y}}_q^+ ({\bar{y}}_q^+-{\bar{y}}_p) (2 (\alpha +{\bar{y}}_p) (3 {\bar{y}}_q^++2 {\bar{y}}_p) +\right. \right. \nonumber \\&\left. \left. +\delta (3 {\bar{y}}_q^++7 {\bar{y}}_p))+({\bar{y}}_q^+-{\bar{y}}_p)^2 \left( \delta (4 \alpha {\bar{y}}_q^+\right. \right. \right. \nonumber \\&\left. \left. \left. -3 \alpha {\bar{y}}_p+{\bar{y}}_q^+ {\bar{y}}_p)+2 {\bar{y}}_q^+ (\alpha +{\bar{y}}_p)^2+2 \delta ^2 {\bar{y}}_q^+\right) \right) +\right. \nonumber \\&\left. +c \delta ^3 {\bar{y}}_p \left( c^2 \left( -6 ({\bar{y}}_q^+)^2-3 {\bar{y}}_q^+ {\bar{y}}_p+{\bar{y}}_p^2\right) \right. \right. \nonumber \\&\left. \left. -c \left( 6 ({\bar{y}}_q^+)^2-7 {\bar{y}}_q^+ {\bar{y}}_p+{\bar{y}}_p^2\right) (\alpha +\delta +{\bar{y}}_p)+\right. \right. \nonumber \\&\left. \left. -({\bar{y}}_q^+-{\bar{y}}_p)^2 \left( 2 \delta ^2+\delta (\alpha +{\bar{y}}_p)+2 (\alpha +{\bar{y}}_p)^2\right) \right) \right) ,\nonumber \\ v_{21}^+=&\frac{2}{3 \delta ^2 {\bar{x}}_t^2 ({\bar{y}}_q^+-{\bar{y}}_p)^4} \left( 6 a^2 {\bar{x}}_t^2 ({\bar{y}}_q^+)^2\right. \nonumber \\&\left. -2 a \delta {\bar{x}}_t {\bar{y}}_q^+ \left[ c (5 {\bar{y}}_q^++{\bar{y}}_p) +({\bar{y}}_q^+-{\bar{y}}_p) (\alpha +\delta +{\bar{y}}_p)\right] +\right. \nonumber \\&\left. +c \delta ^2 \left[ c \left( 3 ({\bar{y}}_q^+)^2+4 {\bar{y}}_q^+ {\bar{y}}_p-{\bar{y}}_p^2\right) \right. \right. \nonumber \\&\left. \left. +(({\bar{y}}_q^+)^2-{\bar{y}}_p^2) (\alpha +\delta +{\bar{y}}_p)\right] \right) ,\nonumber \\ v_{12}^+=&\frac{2 {\bar{y}}_q^+ (c \delta -a {\bar{x}}_t)}{\delta {\bar{x}}_t ({\bar{y}}_q^+-{\bar{y}}_p)^3}, \end{aligned}$$

(A.7)

and the coefficients of the fourth-order terms of the first equation are given by

$$\begin{aligned} p_{31}^+=&\frac{2}{9\delta ^2{\bar{x}}_t^2({\bar{y}}_q^+-{\bar{y}}_p)^5} \left( 16a^2{\bar{x}}_t^2({\bar{y}}_q^+)^2-a\delta {\bar{x}}_t{\bar{y}}_q^+\right. \nonumber \\&\left. \left[ c(17{\bar{y}}_q^++15{\bar{y}}_p)+3({\bar{y}}_q^+-{\bar{y}}_p)(\alpha +\delta +{\bar{y}}_p)\right] +\right. \nonumber \\&\left. +\delta ^2\left[ c^2\left( 3({\bar{y}}_q^+)^2+ 11{\bar{y}}_q^+{\bar{y}}_p+2{\bar{y}}_p^2\right) \right. \right. \nonumber \\&+c({\bar{y}}_q^+-{\bar{y}}_p) ({\bar{y}}_q^++2{\bar{y}}_p)(\alpha +\delta +{\bar{y}}_p)\nonumber \\&\left. \left. -4({\bar{y}}_q^+-{\bar{y}}_p)^2(\alpha +\delta +{\bar{y}}_p)^2\right] \right) ,\nonumber \\ p_{22}^+=&\frac{\delta \left[ c(2{\bar{y}}_q^++{\bar{y}}_p)-({\bar{y}}_q^+-{\bar{y}}_p) (\alpha +\delta +{\bar{y}}_p)\right] -3 a {\bar{x}}_t {\bar{y}}_q^+}{3/2\delta {\bar{x}}_t({\bar{y}}_q^+ -{\bar{y}}_p)^4},\nonumber \\ p_{40}^+=&\frac{-2}{135 \delta ^3 {\bar{x}}_t^3 ({\bar{y}}_q^+-{\bar{y}}_p)^6}\left( 100 a^3 {\bar{x}}_t^3 ({\bar{y}}_q^+)^3\right. \nonumber \\&-3 a^2 \delta {\bar{x}}_t^2 {\bar{y}}_q^+ (20 c {\bar{y}}_q^+ (2 {\bar{y}}_q^++3 {\bar{y}}_p)\nonumber \\&\left. +({\bar{y}}_q^+-{\bar{y}}_p) (20 {\bar{y}}_q^+ (\alpha +{\bar{y}}_p)+\right. \nonumber \\&\left. +\delta (11 {\bar{y}}_q^++9 {\bar{y}}_p)))+3 a \delta ^2 {\bar{x}}_t (c^2 {\bar{y}}_q^+ \left( 9 ({\bar{y}}_q^+)^2\right. \right. \nonumber \\&\left. \left. +62 {\bar{y}}_q^+ {\bar{y}}_p+29 {\bar{y}}_p^2\right) +c ({\bar{y}}_q^+-{\bar{y}}_p) (2 {\bar{y}}_q^+ (\alpha \right. \nonumber \\&\left. +{\bar{y}}_p) (7 {\bar{y}}_q^++13 {\bar{y}}_p)+\right. \nonumber \\&\left. +\delta (8 ({\bar{y}}_q^+)^2+29 {\bar{y}}_q^+ {\bar{y}}_p+3 {\bar{y}}_p^2))-({\bar{y}}_q^+-{\bar{y}}_p)^2 \right. \nonumber \\&(\delta (2 \alpha {\bar{y}}_q^++9 \alpha {\bar{y}}_p+8 {\bar{y}}_q^+ {\bar{y}}_p+3 {\bar{y}}_p^2)\nonumber \\&\left. +{\bar{y}}_q^+ (\alpha +{\bar{y}}_p)^2+\right. \nonumber \\&\left. +\delta ^2 (3 {\bar{y}}_p-2 {\bar{y}}_q^+)))+\delta ^3 (c^3 (-{\bar{y}}_p) (9 {\bar{y}}_q^++{\bar{y}}_p) \right. \nonumber \\&(3 {\bar{y}}_q^++7 {\bar{y}}_p)+3 \alpha ({\bar{y}}_q^+-{\bar{y}}_p) \nonumber \\&\left. (-2 c^2 {\bar{y}}_p (7 {\bar{y}}_q^++3 {\bar{y}}_p)+\right. \nonumber \\&\left. +c {\bar{y}}_p ({\bar{y}}_q^+-{\bar{y}}_p) (11 \delta +2 {\bar{y}}_p)+({\bar{y}}_q^+-{\bar{y}}_p)^2 \right. \nonumber \\&(19 \delta ^2+22 {\bar{y}}_p^2+38 \delta {\bar{y}}_p))-6 c^2 {\bar{y}}_p ({\bar{y}}_q^+-{\bar{y}}_p) \nonumber \\&\left. (7 {\bar{y}}_q^++3 {\bar{y}}_p) (\delta +{\bar{y}}_p)+\right. \nonumber \\&\left. +3 \alpha ^2 ({\bar{y}}_q^+-{\bar{y}}_p)^2 (c {\bar{y}}_p+({\bar{y}}_q^+-{\bar{y}}_p) (19 \delta \right. \nonumber \\&+22 {\bar{y}}_p))+3 c {\bar{y}}_p ({\bar{y}}_q^+-{\bar{y}}_p)^2 (\delta ^2+{\bar{y}}_p^2+11 \delta {\bar{y}}_p)\nonumber \\&\left. +22 \alpha ^3 ({\bar{y}}_q^+-{\bar{y}}_p)^3+\right. \nonumber \\&\left. +({\bar{y}}_q^+-{\bar{y}}_p)^3 (\delta +{\bar{y}}_p) (22 \delta ^2+22 {\bar{y}}_p^2+35 \delta {\bar{y}}_p))\right) . \end{aligned}$$

(A.8)

The computations for the map $P_-$ are totally similar. We use again the formula given in (A.1), this time with

$$\begin{aligned} A^{-}= & {} \begin{bmatrix} -\alpha - N_{RT}{\bar{y}}_p &{}-N_{RT}{\bar{x}}_t&{}0\\ -N_{PI}a &{} -c &{}N_{PI}a\\ \delta -\alpha &{}0&{}-\delta \\ \end{bmatrix},\;\;\widetilde{{\mathbf {x}}}_1=\begin{bmatrix} {\widetilde{x}}_1\\ {\widetilde{y}}_1\\ 0 \end{bmatrix},\\ {\mathbf {v}}^{-}= & {} \begin{bmatrix} {\bar{x}}_t({\bar{y}}_q^+ - N_{RT}{\bar{y}}_p)\\ \frac{a{\bar{x}}_t{\bar{y}}_q^+N_{PI}}{\delta }-c{\bar{y}}_p\\ 0 \end{bmatrix}, \end{aligned}$$

being ${\widetilde{x}}_1\le 0$, $A^-=D{\mathbf {F}}_{-}({\bar{x}}_t,{\bar{y}}_p,C_T)$ and ${\mathbf {v}}^{-}={\mathbf {F}}_{-}({\bar{x}}_t,{\bar{y}}_p,C_T)$. We compute the approximation up to fifth order of the half-return time $t_-=t_-({\widetilde{x}}_1,{\widetilde{y}}_1)$. This is done by solving the equation ${\widetilde{z}}(t_-)=0$ so that, after the evaluation at such a time for the other two coordinates of the solution, we determine the image of the half-return map $({\widetilde{x}}_2,{\widetilde{y}}_2)=P_-({\widetilde{x}}_1,{\widetilde{y}}_1)$, namely

$$\begin{aligned} {\widetilde{x}}_2&=-{\widetilde{x}}_1+p_{20}^-{\widetilde{x}}_1^2+ p_{30}^-{\widetilde{x}}_1^3+p_{21}^-{\widetilde{x}}_1^2{\widetilde{y}}_1+ p_{40}^-{\widetilde{x}}_1^4\nonumber \\&\quad +p_{31}^-{\widetilde{x}}_1^3{\widetilde{y}}_1 +p_{22}^-{\widetilde{x}}_1^2{\widetilde{y}}_1^2+{\widetilde{x}}_1^2 \cdot O(3), \end{aligned}$$

(A.9)

$$\begin{aligned} {\widetilde{y}}_2&=v_{10}^-{\widetilde{x}}_1+{\widetilde{y}}_1+v_{11}^- {\widetilde{x}}_1{\widetilde{y}}_1+v_{20}^-{\widetilde{x}}_1^2+v_{30}^- {\widetilde{x}}_1^3\nonumber \\&\quad +v_{21}^-{\widetilde{x}}_1^2{\widetilde{y}}_1+v_{12}^- {\widetilde{x}}_1{\widetilde{y}}_1^2+{\widetilde{x}}_1\cdot O(3), \end{aligned}$$

(A.10)

with coefficients of the quadratic terms given by

$$\begin{aligned} p_{20}^-= & {} \dfrac{\delta \left[ c{\bar{y}}_p+({\bar{y}}_p-{\bar{y}}_q^-) (\alpha +\delta +N_{RT}{\bar{y}}_p)\right] -a{\bar{x}}_t{\bar{y}}_q^-N_{PI} N_{RT}}{3/2{\bar{x}}_t\delta ({\bar{y}}_q^--{\bar{y}}_p)^2 N_{RT}} ,\nonumber \\ v_{10}^-= & {} \dfrac{2\left( \frac{c\delta }{N_{RT}} {\bar{y}}_p-a{\bar{x}}_t{\bar{y}}_q^-N_{PI}\right) }{{\bar{x}}_t\delta ({\bar{y}}_q^--{\bar{y}}_p)},\nonumber \\ v_{11}^-= & {} \dfrac{2{\bar{y}}_q^-(c\delta -a{\bar{x}}_tN_{PI}N_{RT})}{{\bar{x}}_t \delta ({\bar{y}}_q^--{\bar{y}}_p)^2N_{RT}},\nonumber \\ v_{20}^-= & {} \dfrac{2(aN_{PI}N_{RT}{\bar{x}}_t{\bar{y}}_q^--c\delta {\bar{y}}_p) (2aN_{PI}N_{RT}{\bar{x}}_t{\bar{y}}_q^-+\delta [c({\bar{y}}_p-3{\bar{y}}_q^-) +({\bar{y}}_p-{\bar{y}}_q^-)(\alpha +\delta +N_{RT}{\bar{y}}_p)])}{3{\bar{x}}_t^2\delta ^2({\bar{y}}_q^--{\bar{y}}_p)^3N_{RT}^2}, \end{aligned}$$

(A.11)

coefficients of the cubic terms given by

$$\begin{aligned} p_{21}^-=&\dfrac{\delta \left[ c({\bar{y}}_p+{\bar{y}}_q^-) +({\bar{y}}_p-{\bar{y}}_q^-)(\alpha +\delta +N_{RT}{\bar{y}}_p)\right] -2a{\bar{x}}_t{\bar{y}}_q^-N_{PI}}{3/2{\bar{x}}_t\delta ({\bar{y}}_q^--{\bar{y}}_p)^3 N_{RT}} ,\nonumber \\ p_{30}^-=&\frac{2}{9 \delta ^2 N_{RT}^2 {\bar{x}}_t^2 ({\bar{y}}_q^--{\bar{y}}_p)^4} \left( 4 a^2 N_{PI}^2 N_{RT}^2 {\bar{x}}_t^2 ({\bar{y}}_q^-)^2\right. \nonumber \\&\left. -a \delta N_{PI} N_{RT} {\bar{x}}_t {\bar{y}}_q^- (c (3 {\bar{y}}_q^-+5 {\bar{y}}_p)+\right. \nonumber \\&\left. +({\bar{y}}_q^--{\bar{y}}_p) (\alpha +\delta +N_{RT} {\bar{y}}_p))-\delta ^2 \right. \nonumber \\&\left. \left( -c^2 {\bar{y}}_p (3 {\bar{y}}_q^-+{\bar{y}}_p)+\alpha ({\bar{y}}_p-{\bar{y}}_q^-) (c {\bar{y}}_p\right. \right. \nonumber \\&\left. \left. -4 ({\bar{y}}_q^--{\bar{y}}_p) (\delta +N_{RT} {\bar{y}}_p))+\right. \right. \nonumber \\&+c {\bar{y}}_p ({\bar{y}}_p-{\bar{y}}_q^-) (\delta +N_{RT} {\bar{y}}_p)+2 \alpha ^2 ({\bar{y}}_q^--{\bar{y}}_p)^2\nonumber \\&\left. \left. +2({\bar{y}}_q^--{\bar{y}}_p)^2 (\delta +N_{RT} {\bar{y}}_p)^2\right) \right) ,\nonumber \\ v_{30}^-=&\frac{-2}{9 \delta ^3 N_{RT}^3 {\bar{x}}_t^3 ({\bar{y}}_q^--{\bar{y}}_p)^5} \left( 8 a^3 N_{PI}^3 N_{RT}^3 {\bar{x}}_t^3 ({\bar{y}}_q^-)^3\right. \nonumber \\&\left. -a^2 \delta N_{PI}^2 N_{RT}^2 {\bar{x}}_t^2 {\bar{y}}_q^- (3 c {\bar{y}}_q^- (5 {\bar{y}}_q^-+3 {\bar{y}}_p)+\right. \nonumber \\&+({\bar{y}}_q^--{\bar{y}}_p) (5 \alpha {\bar{y}}_q^-+5 N_{RT} {\bar{y}}_q^- {\bar{y}}_p+2 \delta {\bar{y}}_q^-\nonumber \\&\left. +3 \delta {\bar{y}}_p))+a \delta ^2 N_{PI} N_{RT} {\bar{x}}_t \left( 6 c^2 ({\bar{y}}_q^-)^2 ({\bar{y}}_q^-+3 {\bar{y}}_p)+\right. \right. \nonumber \\&+c {\bar{y}}_q^- ({\bar{y}}_q^--{\bar{y}}_p) (2 (3 {\bar{y}}_q^-+2 {\bar{y}}_p) (\alpha +N_{RT} {\bar{y}}_p)\nonumber \\&\left. \left. +\delta (3 {\bar{y}}_q^-+7 {\bar{y}}_p))+({\bar{y}}_q^--{\bar{y}}_p)^2 \left( \delta (4 \alpha {\bar{y}}_q^-\right. \right. \right. \nonumber \\&\left. \left. \left. -3 \alpha {\bar{y}}_p+N_{RT} {\bar{y}}_q^- {\bar{y}}_p)+\right. \right. \right. \nonumber \\&\left. \left. +2 {\bar{y}}_q^- (\alpha +N_{RT} {\bar{y}}_p)^2+2 \delta ^2 {\bar{y}}_q^-\right) \right) \nonumber \\&\left. +c \delta ^3 {\bar{y}}_p \left( c^2 \left( -6 ({\bar{y}}_q^-)^2-3 {\bar{y}}_q^- {\bar{y}}_p+{\bar{y}}_p^2\right) +\right. \right. \nonumber \\&-c \left( 6 ({\bar{y}}_q^-)^2-7 {\bar{y}}_q^- {\bar{y}}_p+{\bar{y}}_p^2\right) (\alpha +\delta +N_{RT} {\bar{y}}_p)\nonumber \\&\left. \left. -({\bar{y}}_q^--{\bar{y}}_p)^2 \left( 2 \delta ^2+\delta (\alpha +N_{RT} {\bar{y}}_p)\right. \right. \right. \nonumber \\&\left. \left. \left. +2 (\alpha +N_{RT} {\bar{y}}_p)^2\right) \right) \right) ,\nonumber \\ v_{21}^-=&\frac{2}{3 \delta ^2 N_{RT}^2 {\bar{x}}_t^2 ({\bar{y}}_q^--{\bar{y}}_p)^4} \left( 6 a^2 N_{PI}^2 N_{RT}^2 {\bar{x}}_t^2 ({\bar{y}}_q^-)^2\right. \nonumber \\&-2 a \delta N_{PI} N_{RT} {\bar{x}}_t {\bar{y}}_q^- (c (5 {\bar{y}}_q^-+{\bar{y}}_p)+\nonumber \\&+({\bar{y}}_q^--{\bar{y}}_p) (\alpha +\delta +N_{RT} {\bar{y}}_p))\nonumber \\&+c \delta ^2 \left( c \left( 3 ({\bar{y}}_q^-)^2+4 {\bar{y}}_q^- {\bar{y}}_p-{\bar{y}}_p^2\right) \right. \nonumber \\&\left. \left. +({\bar{y}}_q^--{\bar{y}}_p) ({\bar{y}}_q^-+{\bar{y}}_p) (\alpha +\delta +N_{RT} {\bar{y}}_p)\right) \right) ,\nonumber \\ v_{12}^-=&\frac{2 {\bar{y}}_q^- (c \delta -a N_{PI} N_{RT} {\bar{x}}_t)}{\delta N_{RT} {\bar{x}}_t ({\bar{y}}_q^--{\bar{y}}_p)^3}, \end{aligned}$$

(A.12)

and the coefficients of the fourth-order terms of the first equation are given by

$$\begin{aligned} p_{31}^-=&\frac{2}{9 \delta ^2 N_{RT}^2 {\bar{x}}_t^2 ({\bar{y}}_q^--{\bar{y}}_p)^5} \left( 16 a^2 N_{PI}^2 N_{RT}^2 {\bar{x}}_t^2 ({\bar{y}}_q^-)^2\right. \nonumber \\&-a \delta N_{PI} N_{RT} {\bar{x}}_t {\bar{y}}_q^- (c (17 {\bar{y}}_q^-+15 {\bar{y}}_p)+\nonumber \\&+3 ({\bar{y}}_q^--{\bar{y}}_p) (\alpha +\delta +N_{RT} {\bar{y}}_p))\nonumber \\&\left. +\delta ^2 \left( c^2 \left( 3 ({\bar{y}}_q^-)^2+11 {\bar{y}}_q^- {\bar{y}}_p+2 {\bar{y}}_p^2\right) +\right. \right. \nonumber \\&+c ({\bar{y}}_q^--{\bar{y}}_p) ({\bar{y}}_q^-+2 {\bar{y}}_p) (\alpha +\delta +N_{RT} {\bar{y}}_p)\nonumber \\&\left. \left. -4 ({\bar{y}}_q^--{\bar{y}}_p)^2 (\alpha +\delta +N_{RT} {\bar{y}}_p)^2\right) \right) ,\nonumber \\ p_{22}^-=&\frac{ \delta \left[ c (2 {\bar{y}}_q^-+{\bar{y}}_p)-({\bar{y}}_q^--{\bar{y}}_p) (\alpha +\delta +N_{RT} {\bar{y}}_p)\right] -3 a N_{PI} N_{RT} {\bar{x}}_t {\bar{y}}_q^-}{3/2 \delta N_{RT} {\bar{x}}_t ({\bar{y}}_q^--{\bar{y}}_p)^4},\nonumber \\ p_{40}^-=&\frac{-2}{135 \delta ^3 N_{RT}^3 {\bar{x}}_t^3 ({\bar{y}}_q^--{\bar{y}}_p)^6} \left( 100 a^3 N_{PI}^3 N_{RT}^3 {\bar{x}}_t^3 ({\bar{y}}_q^-)^3\right. \nonumber \\&-3 a^2 \delta N_{PI}^2 N_{RT}^2 {\bar{x}}_t^2 {\bar{y}}_q^- (20 c {\bar{y}}_q^- (2 {\bar{y}}_q^-+3 {\bar{y}}_p)+\nonumber \\&+({\bar{y}}_q^--{\bar{y}}_p) (20 {\bar{y}}_q^- (\alpha +N_{RT} {\bar{y}}_p)+\delta (11 {\bar{y}}_q^-+9 {\bar{y}}_p)))\nonumber \\&\left. -3 a \delta ^2 N_{PI} N_{RT} {\bar{x}}_t \left( -c^2 {\bar{y}}_q^- \left( 9 ({\bar{y}}_q^-)^2+62 {\bar{y}}_q^- {\bar{y}}_p+29 {\bar{y}}_p^2\right) +\right. \right. \nonumber \\&+c (-({\bar{y}}_q^--{\bar{y}}_p)) \left( 2 {\bar{y}}_q^- (7 {\bar{y}}_q^-+13 {\bar{y}}_p) (\alpha +N_{RT} {\bar{y}}_p)\right. \nonumber \\&\left. \left. \left. +\delta \left( 8 ({\bar{y}}_q^-)^2+29 {\bar{y}}_q^- {\bar{y}}_p+3 {\bar{y}}_p^2\right) \right) +\right. \right. \nonumber \\&+({\bar{y}}_q^--{\bar{y}}_p)^2 \left( \delta (\alpha (2 {\bar{y}}_q^-+9 {\bar{y}}_p)+N_{RT} {\bar{y}}_p (8 {\bar{y}}_q^-\right. \nonumber \\&\left. \left. \left. +3 {\bar{y}}_p))+{\bar{y}}_q^- (\alpha +N_{RT} {\bar{y}}_p)^2+\delta ^2 (3 {\bar{y}}_p-2 {\bar{y}}_q^-)\right) \right) +\right. \nonumber \\&+\delta ^3 \left( c^3 (-{\bar{y}}_p) (9 {\bar{y}}_q^-+{\bar{y}}_p) (3 {\bar{y}}_q^-+7 {\bar{y}}_p)\right. \nonumber \\&\left. \left. +3 \alpha ({\bar{y}}_q^--{\bar{y}}_p) \left( -2 c^2 {\bar{y}}_p (7 {\bar{y}}_q^-+3 {\bar{y}}_p)+\right. \right. \right. \nonumber \\&+c {\bar{y}}_p ({\bar{y}}_q^--{\bar{y}}_p) (11 \delta +2 N_{RT} {\bar{y}}_p)+({\bar{y}}_q^--{\bar{y}}_p)^2 \nonumber \\&\left. \left. \left. \left( 19 \delta ^2+22 N_{RT}^2 {\bar{y}}_p^2+38 \delta N_{RT} {\bar{y}}_p\right) \right) +\right. \right. \nonumber \\&-6 c^2 {\bar{y}}_p ({\bar{y}}_q^--{\bar{y}}_p) (7 {\bar{y}}_q^-+3 {\bar{y}}_p) (\delta +N_{RT} {\bar{y}}_p)\nonumber \\&\left. \left. +3 \alpha ^2 ({\bar{y}}_q^--{\bar{y}}_p)^2 (c {\bar{y}}_p+({\bar{y}}_q^--{\bar{y}}_p) (19 \delta +22 N_{RT} {\bar{y}}_p))+\right. \right. \nonumber \\&\left. \left. +3 c {\bar{y}}_p ({\bar{y}}_q^--{\bar{y}}_p)^2 \left( \delta ^2+N_{RT}^2 {\bar{y}}_p^2+11 \delta N_{RT} {\bar{y}}_p\right) +\right. \right. \nonumber \\&+22 \alpha ^3 ({\bar{y}}_q^--{\bar{y}}_p)^3+({\bar{y}}_q^--{\bar{y}}_p)^3 (\delta +N_{RT} {\bar{y}}_p) \nonumber \\&\left. \left. \left( 22 \delta ^2+22 N_{RT}^2 {\bar{y}}_p^2+35 \delta N_{RT} {\bar{y}}_p\right) \right) \right) . \end{aligned}$$

(A.13)

Remark 5

Substituting (36) in $v_{10}^-$ and $v_{10}^+$, we prove that $v_{10}^-=v_{10}^+$.

Remark 6

Although in the half-return maps $P_-$ and $P_+$ we have not given the coefficients of the fifth-order terms of the first equation and the fourth and fifth-order terms of the second equation, such coefficients can be calculated since it appears in terms of the third and fourth order of the determinant and trace given in (47). We prefer not to show such coefficients since they are not necessary for the conclusion of this study.