Appendix: The Second Order Averaging Function for Planar Piecewise Systems
In this appendix we present the formula and its derivation of the associated second order averaging function for the planar piecewise smooth differential system (8).
Continuing the computations given in the proof of Theorem 3, we get
$$\begin{aligned} \begin{aligned} \frac{\partial F_1^\pm (\theta ,r)}{\partial r}&=\frac{g_0^\pm (f_1^\pm )'-f_0^\pm (g_1^\pm )'-(f_0^\pm )'g_1^\pm -f_1^\pm (g_0^\pm )'}{(g_0^\pm )^2}+\frac{2f_0^\pm g_1^\pm (g_0^\pm )'}{(g_0^\pm )^3}\\&=\frac{1}{\big ((\xi _1^\pm )'\eta _1^\pm q_0^\pm -(\xi _2^\pm )'\eta _2^\pm p_0^\pm \big )^2}\Bigg ((\xi ^\pm _1)'(\xi ^\pm _2)'((\eta ^\pm _1)'\eta ^\pm _2-(\eta ^\pm _2)'\eta ^\pm _1)(p_0^\pm q_1^\pm \\&\quad -q_0^\pm p_1^\pm ) -((\xi ^\pm _1)''\xi ^\pm _2\eta ^\pm _1(\eta ^\pm _2)'+(\xi ^\pm _2)''\xi ^\pm _1\eta ^\pm _2(\eta ^\pm _1)')(p_0^\pm q_1^\pm +q_0^\pm p_1^\pm )\\&\quad +2((\xi ^\pm _1)''\xi ^\pm _1\eta ^\pm _1(\eta ^\pm _1)'q_0^\pm q_1^\pm +(\xi ^\pm _2)''\xi ^\pm _2\eta ^\pm _2(\eta ^\pm _2)'p_0^\pm p_1^\pm )+\big (\xi ^\pm _1(\xi ^\pm _2)'(\eta ^\pm _1)'\eta ^\pm _2-\\&\quad (\xi ^\pm _1)'\xi ^\pm _2\eta ^\pm _1(\eta ^\pm _2)'\big )\bigg ((\xi _1^\pm )'\Big (p_0^\pm \frac{\partial q_1^\pm }{\partial x}-q_0^\pm \frac{\partial p_1^\pm }{\partial x}\Big )+(\xi _2^\pm )'\Big (p_0^\pm \frac{\partial q_1^\pm }{\partial y}-q_0^\pm \frac{\partial p_1^\pm }{\partial y}\Big )\bigg )\\&\quad -\big (\xi ^\pm _1(\xi ^\pm _2)'(\eta ^\pm _1)'\eta ^\pm _2+(\xi ^\pm _1)'\xi ^\pm _2\eta ^\pm _1(\eta ^\pm _2)'\big )\bigg ((\xi _1^\pm )'\Big (p_1^\pm \frac{\partial q_0^\pm }{\partial x}+q_1^\pm \frac{\partial p_0^\pm }{\partial x}\Big )\\&\quad +(\xi _2^\pm )'\Big (p_1^\pm \frac{\partial q_0^\pm }{\partial y}+q_1^\pm \frac{\partial p_0^\pm }{\partial y}\Big )\bigg )+2(((\xi ^\pm _1)')^2\xi ^\pm _1\eta ^\pm _1(\eta ^\pm _1)'q_1^\pm \frac{\partial q_0^\pm }{\partial x}\\&\quad +(\xi ^\pm _2)'(\xi ^\pm _1)'\xi ^\pm _2\eta ^\pm _2(\eta ^\pm _2)'p_1^\pm \frac{\partial p_0^\pm }{\partial x})+2((\xi ^\pm _1)'(\xi ^\pm _2)'\xi ^\pm _1\eta ^\pm _1(\eta ^\pm _1)'q_1^\pm \frac{\partial q_0^\pm }{\partial y}+\\&\quad ((\xi ^\pm _2)')^2\xi ^\pm _2\eta ^\pm _2(\eta ^\pm _2)'p_1^\pm \frac{\partial p_0^\pm }{\partial y}) \Bigg )+\frac{1}{\big ((\xi _1^\pm )'\eta _1^\pm q_0^\pm -(\xi _2^\pm )'\eta _2^\pm p_0^\pm \big )^3}\bigg (\big (\xi _2^\pm (\eta _2^\pm )'p_0^\pm \\&\quad -\xi _1^\pm (\eta _1^\pm )'q_0^\pm \big )\big ((\xi _1^\pm )'\eta _1^\pm q_1^\pm -(\xi _2^\pm )'\eta _2^\pm p_1^\pm \big )\Big ((\xi _1^\pm )''\eta _1^\pm q_1^\pm -(\xi _2^\pm )''\eta _2^\pm p_1^\pm \\&\quad +(\xi _1^\pm )'\eta _1^\pm (\xi _1^\pm \frac{\partial q_0^\pm }{\partial x}+\xi _2^\pm \frac{\partial q_0^\pm }{\partial y})-(\xi _2^\pm )'\eta _2^\pm (\xi _1^\pm \frac{\partial p_0^\pm }{\partial x}+\xi _2^\pm \frac{\partial p_0^\pm }{\partial y})\Big )\bigg ), \end{aligned} \end{aligned}$$
(54)
and
$$\begin{aligned} \frac{\partial ^2 F_0^\pm (\theta ,r)}{\partial r^2}= & {} \frac{1}{\big (g_0^\pm \big )^3}\left( (g_0^\pm )^2\frac{\partial ^2}{\partial r^2}f_0^\pm -g_0^\pm f_0^\pm \frac{\partial ^2}{\partial r^2}g_0^\pm -2g_0^\pm \frac{\partial }{\partial r}g_0^\pm \frac{\partial }{\partial r}f_0^\pm +2f_0^\pm (\frac{\partial }{\partial r}g_0^\pm )^2\right) \nonumber \\= & {} \frac{1}{\big ((\xi _1^\pm )'\eta _1^\pm q_0^\pm -(\xi _2^\pm )'\eta _2^\pm p_0^\pm \big )^3}\Bigg (\bigg (\big ((\xi _1^\pm )'\big )^2(\xi _1^\pm )''-2\xi _1^\pm \big ((\xi _1^\pm )''\big )^2\nonumber \\&\quad +\,\xi _1^\pm (\xi _1^\pm )'(\xi _1^\pm )'''\bigg )\big (\eta _1^\pm \big )^2(\eta _1^\pm )'\big (q_0^\pm \big )^3+\bigg (-\big ((\xi _2^\pm )'\big )^2(\xi _2^\pm )''+2\xi _2^\pm \big ((\xi _2^\pm )''\big )^2\nonumber \\&\quad -\,\xi _2^\pm (\xi _2^\pm )'(\xi _2^\pm )'''\bigg )\big (\eta _2^\pm \big )^2(\eta _2^\pm )'\big (p_0^\pm \big )^3+\bigg (\Big (\xi _1^\pm (\xi _2^\pm )'(\xi _2^\pm )'''-(\xi _1^\pm )''((\xi _2^\pm )')^2\nonumber \\&\quad +\,2(\xi _1^\pm )'(\xi _2^\pm )'(\xi _2^\pm )''-2\xi _1^\pm ((\xi _2^\pm )'')^2\Big )(\eta _1^\pm )'(\eta _2^\pm \big )^2+\Big (2(\xi _1^\pm )''((\xi _2^\pm )')^2\nonumber \\&\quad +\,(\xi _1^\pm )'\xi _2^\pm (\xi _2^\pm )'''+(\xi _1^\pm )'''\xi _2^\pm (\xi _2^\pm )'-4(\xi _1^\pm )''\xi _2^\pm (\xi _2^\pm )''\Big )\eta _1^\pm \eta _2^\pm (\eta _2^\pm )'\bigg )(p_0^\pm \big )^2q_0^\pm \nonumber \\&\quad +\,\bigg (\big (-(\xi _1^\pm )'(\xi _1^\pm )'''\xi _2^\pm +((\xi _1^\pm )')^2(\xi _2^\pm )''-2(\xi _1^\pm )'(\xi _1^\pm )''(\xi _2^\pm )'\nonumber \\&\quad +\,2((\xi _1^\pm )'')^2\xi _2^\pm \big )(\eta _1^\pm \big )^2(\eta _2^\pm )'+\big (-2((\xi _1^\pm )')^2(\xi _2^\pm )''-\xi _1^\pm (\xi _1^\pm )'''(\xi _2^\pm )'\nonumber \\&\quad -\,\xi _1^\pm (\xi _1^\pm )'(\xi _2^\pm )'''+4\xi _1^\pm (\xi _1^\pm )''(\xi _2^\pm )''\big )\eta _1^\pm \eta _2^\pm (\eta _2^\pm )'\bigg )(q_0^\pm \big )^2p_0^\pm \nonumber \\&\quad +\,\Bigg (2\big ((\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )' - \xi _1^\pm (\xi _2^\pm )' (\eta _1^\pm )'\eta _2^\pm \big ) \big ((\xi _1^\pm )'\big )^2 (\xi _2^\pm )' \eta _1^\pm \frac{\partial ^2 p_0^\pm }{\partial x\partial y}\nonumber \\&\quad (\xi _1^\pm )'\eta _1^\pm \left( 3\xi _1^\pm (\xi _2^\pm )'(\eta _1^\pm )'\eta _2^\pm -(\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )'\right) \left( (\xi _1^\pm )''\frac{\partial p_0^\pm }{\partial x}-(\xi _2^\pm )''\frac{\partial p_0^\pm }{\partial y}\right) \nonumber \\ \end{aligned}$$
$$\begin{aligned}&\quad +\,(\xi _1^\pm )' \eta _1^\pm \left( (\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )' -\xi _1^\pm (\xi _2^\pm )' (\eta _1^\pm )' \eta _2^\pm \right) \Big (\big ((\xi _1^\pm )'\big )^2\frac{\partial ^2 p_0^\pm }{\partial x^2}\nonumber \\&\quad +\,\big ((\xi _2^\pm )'\big )^2\frac{\partial ^2 p_0^\pm }{\partial y^2} \Big )-2 \xi _1^\pm \big ((\xi _1^\pm )'\big )^2 (\xi _2^\pm )''\eta _1^\pm (\eta _1^\pm )' \eta _2^\pm \frac{\partial p_0^\pm }{\partial x}\nonumber \\&\quad +\,2(\xi _1^\pm )''(\xi _2^\pm )'\eta _1^\pm \Big ( 2\xi _1^\pm (\xi _2^\pm )'(\eta _1^\pm )' \eta _2^\pm -(\xi _1^\pm )'\xi _2^\pm \eta _1^\pm \big (\eta _2^\pm )') \frac{\partial p_0^\pm }{\partial y}\Bigg )(q_0^\pm )^2\nonumber \\&\quad +\,\Bigg (2\Big ( (\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )' - \xi _1^\pm (\xi _2^\pm )' (\eta _1^\pm )' \eta _2^\pm \Big ) (\xi _1^\pm )'\big ((\xi _2^\pm )'\big )^2\eta _2^\pm \frac{\partial ^2 q_0^\pm }{\partial x\partial y}+\nonumber \\&\quad (\xi _2^\pm )'\eta _2^\pm \left( 3(\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )'-\xi _1^\pm (\xi _2^\pm )'(\eta _1^\pm )'\eta _2^\pm \right) \left( (\xi _1^\pm )''\frac{\partial q_0^\pm }{\partial x}-(\xi _2^\pm )''\frac{\partial q_0^\pm }{\partial y}\right) \nonumber \\&\quad +\,2(\xi _1^\pm )'\big ((\xi _2^\pm )'\big )^2\eta _2^\pm \left( \eta _1^\pm (\eta _2^\pm )'-(\eta _1^\pm )'\eta _2^\pm \right) \left( (\xi _1^\pm )'\frac{\partial q_0^\pm }{\partial x}+(\xi _2^\pm )'\frac{\partial q_0^\pm }{\partial y}\right) \nonumber \\&\quad +\,(\xi _2^\pm )' \eta _2^\pm \left( (\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )' -\xi _1^\pm (\xi _2^\pm )' (\eta _1^\pm )' \eta _2^\pm \right) \Big (\big ((\xi _1^\pm )'\big )^2\frac{\partial ^2 q_0^\pm }{\partial x^2} \nonumber \\&\quad +\,\big ((\xi _2^\pm )'\big )^2\frac{\partial ^2 q_0^\pm }{\partial y^2} \Big )+2(\xi _1^\pm )'(\xi _2^\pm )'' \eta _2^\pm \Big ( \xi _1^\pm (\xi _2^\pm )' (\eta _1^\pm )'\eta _2^\pm \nonumber \\&\quad -\,2 (\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )' \Big ) \frac{\partial q_0^\pm }{\partial x} +2(\xi _1^\pm )''\xi _2^\pm \big ((\xi _2^\pm )'\big )^2\eta _1^\pm \eta _2^\pm (\eta _2^\pm )' \frac{\partial q_0^\pm }{\partial y}\Bigg )(p_0^\pm )^2 \end{aligned}$$
$$\begin{aligned}&\quad +\,\Bigg ((\xi _2^\pm )'\Big ( \xi _1^\pm (\xi _2^\pm )' (\eta _1^\pm )'(\eta _2^\pm )^2 -(\xi _1^\pm )'\xi _2^\pm \eta _1^\pm (\eta _2^\pm )' \eta _2^\pm \Big )\\&\qquad \qquad \times \,\left( \big ((\xi _2^\pm )'\big )^2 \frac{\partial ^2 p_0^\pm }{\partial y^2}+\big ((\xi _1^\pm )'\big )^2 \frac{\partial ^2 p_0^\pm }{\partial x^2}\right) \\&\quad +\,(\xi _1^\pm )'\Big (\xi _1^\pm (\xi _2^\pm )' \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm - (\xi _1^\pm )'\xi _2^\pm (\eta _1^\pm )^2(\eta _2^\pm )' \Big )\\&\qquad \qquad \times \,\left( \big ((\xi _1^\pm )'\big )^2 \frac{\partial ^2 q_0^\pm }{\partial x^2}+\big ((\xi _2^\pm )'\big )^2 \frac{\partial ^2 q_0^\pm }{\partial y^2}\right) \\&\quad +\,2\Big ( \xi _1^\pm (\xi _2^\pm )' (\eta _1^\pm )' \eta _2^\pm - (\xi _1^\pm )' \xi _2^\pm \eta _1^\pm (\eta _2^\pm )' \Big ) \big ((\xi _1^\pm )'\big )^2 (\xi _2^\pm )'\eta _1^\pm \frac{\partial ^2 q_0^\pm }{\partial x\partial y}\\&\quad +\,2\Big ( \xi _1^\pm (\xi _2^\pm )'(\eta _1^\pm )' \eta _2^\pm - (\xi _1^\pm )' \xi _2^\pm \eta _1^\pm (\eta _2^\pm )' \Big )(\xi _1^\pm )'\big ((\xi _2^\pm )'\big )^2\eta _2^\pm \frac{\partial ^2 p_0^\pm }{\partial x\partial y}\\&\quad +\,\Big (2 \big ((\xi _1^\pm )'\big )^3(\xi _2^\pm )' \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm \\&\quad -\,3 \xi _1^\pm (\xi _1^\pm )' (\xi _1^\pm )'' (\xi _2^\pm )' \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm +2 \xi _1^\pm \big ((\xi _1^\pm )'\big )^2(\xi _2^\pm )'' \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm \\&\quad +\,2 \big ((\xi _1^\pm )'\big )^3 (\xi _2^\pm )'(\eta _1^\pm )^2 (\eta _2^\pm )'+\big ((\xi _1^\pm )'\big )^2 (\xi _1^\pm )'' \xi _2^\pm (\eta _1^\pm )^2 (\eta _2^\pm )'\Big ) \frac{\partial q_0^\pm }{\partial x}\\&\quad +\,\Big (3 \xi _1^\pm (\xi _1^\pm )'(\xi _2^\pm )'(\xi _2^\pm )'' \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm +2 \big ((\xi _1^\pm )'\big )^2\big ((\xi _2^\pm )'\big )^2 \eta _1^\pm (\eta _1^\pm )'\eta _2^\pm \\&\quad +\,2 (\xi _1^\pm )' (\xi _1^\pm )''\xi _2^\pm (\xi _2^\pm )' (\eta _1^\pm )^2 (\eta _2^\pm )' -4 \xi _1^\pm (\xi _1^\pm )''\big ((\xi _2^\pm )'\big )^2 \eta _1^\pm \eta _2^\pm \\&\quad -\,2 \big ((\xi _1^\pm )'\big )^2 \big ((\xi _2^\pm )'\big )^2(\eta _1^\pm )^2 (\eta _2^\pm )'-\big ((\xi _1^\pm )'\big )^2 \xi _2^\pm (\xi _2^\pm )''(\eta _1^\pm )^2 (\eta _2^\pm )' \Big ) \frac{\partial q_0^\pm }{\partial y}\\&\quad +\,\Big (-2 \xi _1^\pm (\xi _1^\pm )'(\xi _2^\pm )' (\xi _2^\pm )''(\eta _1^\pm )' (\eta _2^\pm )^2 -3(\xi _1^\pm )'(\xi _1^\pm )''\xi _2^\pm (\xi _2^\pm )' \eta _1^\pm \eta _2^\pm (\eta _2^\pm )'\\ \end{aligned}$$
$$\begin{aligned}&\quad -\,2 \big ((\xi _1^\pm )'\big )^2\big ((\xi _2^\pm )'\big )^2 \eta _1^\pm \eta _2^\pm (\eta _2^\pm )'+4 \big ((\xi _1^\pm )'\big )^2\xi _2^\pm (\xi _2^\pm )''\eta _1^\pm \eta _2^\pm (\eta _2^\pm )' \\&\quad +\,\xi _1^\pm (\xi _1^\pm )''\big ((\xi _2^\pm )'\big )^2 (\eta _1^\pm )' (\eta _2^\pm )^2 +2 \big ((\xi _1^\pm )'\big )^2 \big ((\xi _2^\pm )'\big )^2 (\eta _1^\pm )'(\eta _2^\pm )^2 \Big ) \frac{\partial p_0^\pm }{\partial x}\\&\quad +\,\Big (-2 (\xi _1^\pm )' \big ((\xi _2^\pm )'\big )^3\eta _1^\pm \eta _2^\pm (\eta _2^\pm )'-2 (\xi _1^\pm )''\xi _2^\pm \big ((\xi _2^\pm )'\big )^2 \eta _1^\pm \eta _2^\pm (\eta _2^\pm )'\\&\quad -\,\xi _1^\pm \big ((\xi _2^\pm )'\big )^2 (\xi _2^\pm )''(\eta _1^\pm )' (\eta _2^\pm )^2+3 (\xi _1^\pm )'\xi _2^\pm (\xi _2^\pm )' (\xi _2^\pm )'' \eta _1^\pm \eta _2^\pm (\eta _2^\pm )' \\&\quad +\,2 (\xi _1^\pm )'\big ((\xi _2^\pm )'\big )^3 (\eta _1^\pm )' (\eta _2^\pm )^2 \Big ) \frac{\partial p_0^\pm }{\partial y}\Bigg )p_0^\pm q_0^\pm \\&\quad +\,\Bigg (\Big (-2 \xi _1^\pm \big ((\xi _2^\pm )'\big )^4 (\eta _1^\pm )' (\eta _2^\pm )^2+2 (\xi _1^\pm )' \xi _2^\pm \big ((\xi _2^\pm )'\big )^3 \eta _1^\pm \eta _2^\pm (\eta _2^\pm )'\Big ) \left( \frac{\partial p_0^\pm }{\partial y}\right) ^2\\&\quad +\,2\Big ( (\xi _1^\pm )' \xi _2^\pm (\xi _2^\pm )' \eta _1^\pm \eta _2^\pm (\eta _2^\pm )' -2 \xi _1^\pm \big ((\xi _2^\pm )'\big )^2 (\eta _1^\pm )' (\eta _2^\pm )^2 \Big )\big ((\xi _1^\pm )'\big )^2 \left( \frac{\partial p_0^\pm }{\partial x}\right) ^2\\&\quad +\,4\Big ( \big ((\xi _1^\pm )'\big )^2\xi _2^\pm \big ((\xi _2^\pm )'\big )^2\eta _1^\pm \eta _2^\pm (\eta _2^\pm )' - \xi _1^\pm (\xi _1^\pm )'\big ((\xi _2^\pm )'\big )^3 (\eta _1^\pm )' (\eta _2^\pm )^2 \Big ) \frac{\partial p_0^\pm }{\partial y} \frac{\partial p_0^\pm }{\partial x}\\&\quad +\,2\Big ( \xi _1^\pm \big ((\xi _1^\pm )'\big )^2 \big ((\xi _2^\pm )'\big )^2 \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm - \big ((\xi _1^\pm )'\big )^3 \xi _2^\pm (\xi _2^\pm )' (\eta _1^\pm )^2(\eta _2^\pm )' \Big ) \frac{\partial q_0^\pm }{\partial x} \frac{\partial p_0^\pm }{\partial y} \end{aligned}$$
$$\begin{aligned}&\quad +\,2\Big ( \xi _1^\pm \big ((\xi _1^\pm )'\big )^2\big ((\xi _2^\pm )'\big )^2 \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm -\big ((\xi _1^\pm )'\big )^3 \xi _2^\pm (\xi _2^\pm )' (\eta _1^\pm )^2(\eta _2^\pm )' \Big ) \frac{\partial q_0^\pm }{\partial y} \frac{\partial p_0^\pm }{\partial x}\nonumber \\&\quad +\,2\Big (- \big ((\xi _1^\pm )'\big )^2\xi _2^\pm (\eta _1^\pm )^2 (\eta _2^\pm )' + \xi _1^\pm (\xi _1^\pm )'(\xi _2^\pm )' \eta _1^\pm (\eta _1^\pm )' \eta _2^\pm \Big )\big ((\xi _2^\pm )'\big )^2 \frac{\partial q_0^\pm }{\partial y} \frac{\partial p_0^\pm }{\partial y}\nonumber \\&\quad +\,2\Big (- \big ((\xi _1^\pm )'\big )^4 \xi _2^\pm (\eta _1^\pm )^2(\eta _2^\pm )' +2 \xi _1^\pm \big ((\xi _1^\pm )'\big )^3(\xi _2^\pm )' \eta _1^\pm (\eta _1^\pm )'\eta _2^\pm \Big ) \frac{\partial q_0^\pm }{\partial x} \frac{\partial p_0^\pm }{\partial x}\Bigg )q_0^\pm \nonumber \\&\quad +\,\Bigg (2\Big ( (\xi _1^\pm )'\xi _2^\pm (\eta _1^\pm )^2(\eta _2^\pm )' -\xi _1^\pm (\xi _2^\pm )'\eta _1^\pm (\eta _1^\pm )' \eta _2^\pm \Big ) \big ((\xi _1^\pm )'\big )^3\left( \frac{\partial q_0^\pm }{\partial x}\right) ^2\nonumber \\&\quad +\,2\Big ( \big ((\xi _1^\pm )'\big )^2\xi _2^\pm (\eta _1^\pm )^2 (\eta _2^\pm )' - \xi _1^\pm (\xi _1^\pm )' (\xi _2^\pm )' \eta _1^\pm (\eta _1^\pm )'\eta _2^\pm \Big )\big ((\xi _2^\pm )'\big )^2 \left( \frac{\partial q_0^\pm }{\partial y}\right) ^2\nonumber \\&\quad +\,2\Big (- \big ((\xi _1^\pm )'\big )^2\xi _2^\pm \eta _1^\pm \eta _2^\pm (\eta _2^\pm )' + \xi _1^\pm (\xi _1^\pm )'(\xi _2^\pm )'(\eta _1^\pm )' (\eta _2^\pm )^2 \Big ) \big ((\xi _2^\pm )'\big )^2\frac{\partial q_0^\pm }{\partial x} \frac{\partial p_0^\pm }{\partial y}\nonumber \\&\quad +\,2\Big (- \big ((\xi _1^\pm )'\big )^2\xi _2^\pm \eta _1^\pm \eta _2^\pm (\eta _2^\pm )'+ \xi _1^\pm (\xi _1^\pm )'(\xi _2^\pm )'(\eta _1^\pm )' (\eta _2^\pm )^2 \Big ) \big ((\xi _2^\pm )'\big )^2\frac{\partial q_0^\pm }{\partial y} \frac{\partial p_0^\pm }{\partial x}\nonumber \\&\quad +\,2\Big (- (\xi _1^\pm )' \xi _2^\pm \eta _1^\pm \eta _2^\pm (\eta _2^\pm )' + \xi _1^\pm (\eta _1^\pm )' (\xi _2^\pm )' (\eta _2^\pm )^2\Big ) \big ((\xi _2^\pm )'\big )^3\frac{\partial q_0^\pm }{\partial y} \frac{\partial p_0^\pm }{\partial y}\nonumber \\&\quad +\,2\Big ( \xi _1^\pm \big ((\xi _2^\pm )'\big )^2 (\eta _1^\pm )' (\eta _2^\pm )^2 - (\xi _1^\pm )' \xi _2^\pm (\xi _2^\pm )' \eta _1^\pm \eta _2^\pm (\eta _2^\pm )' \Big ) \big ((\xi _1^\pm )'\big )^2\frac{\partial q_0^\pm }{\partial x} \frac{\partial p_0^\pm }{\partial x}\nonumber \\&\quad +\,4\Big ( (\xi _1^\pm )'\xi _2^\pm (\eta _1^\pm )^2 (\eta _2^\pm )' - \xi _1^\pm (\xi _2^\pm )' \eta _1^\pm (\eta _1^\pm )'\eta _2^\pm \Big ) (\xi _2^\pm )'\big ((\xi _1^\pm )'\big )^2\frac{\partial q_0^\pm }{\partial y} \frac{\partial q_0^\pm }{\partial x}\Bigg )p_0^\pm . \end{aligned}$$
(55)
Set
$$\begin{aligned} v_1^1(\theta ,z) =&\int _0^\theta \frac{(-\widetilde{p}_1^+ \widetilde{q}_0^++\widetilde{p}_0^+ \widetilde{q}_1^+)}{\frac{\partial r_0^+(\theta ,z)}{\partial z}\Big ((\xi _1^+)'(r_0^+)\eta _1^+ \widetilde{q}_0^+-(\xi _2^+)'(r_0^+)\eta _2^+ \widetilde{p}_0^+\Big )^2}\\&\times \Big (\xi _1^+(r_0^+)(\xi _2^+)'(r_0^+)(\eta _1^+)'\eta _2^+-(\xi _1^+)' (r_0^+)\xi _2^+(r_0^+)\eta _1^+(\eta _2^+)'\Big )d\theta ,\\&v_1^2(\theta ,z)\\ =&v_1^1(\pi ,z)+\int _\pi ^{\theta }\frac{(-\widetilde{p}_1^- \widetilde{q}_0^-+\widetilde{p}_0^- \widetilde{q}_1^-)}{\frac{\partial r_0^-(\theta ,z)}{\partial z}\Big ((\xi _1^-)'(r_0^-)\eta _1^- \widetilde{q}_0^--(\xi _2^-)'(r_0^-)\eta _2^- \widetilde{p}_0^-\Big )^2}\\&\times \Big (\xi _1^-(\xi _2^-)'(\eta _1^-)'(r_0^-)\eta _2(r_0^-(\theta ,z)) -(\xi _1^-)'(r_0^-)\xi _2^-(r_0^-)\eta _1^-(\eta _2^-)'\Big )d\theta . \end{aligned}$$
From the expression (6), we have
$$\begin{aligned} \begin{aligned} H_2(z)&=\displaystyle \int \limits _0^{\pi }\frac{1}{\frac{\partial r_0^+(s,z)}{\partial z}}\left( F^+_2(s,r^+_0(s,z))+\frac{\partial F^+_1}{\partial r}(s,r^+_0(s,z))r^+_1(s,z)\right. \\ \quad&\qquad \qquad \qquad \qquad \left. +\frac{1}{2}\frac{\partial ^2 F^+_0}{\partial r^2}(s,x_0(s,z))(r^+_1(s,z))^2\right) ds\\&\qquad +\displaystyle \int \limits _\pi ^{2\pi }\frac{1}{\frac{\partial r_0^-(s,z)}{\partial z}}\left( F^-_2(s,r^-_0(s,z))+\frac{\partial F^-_1}{\partial r}(s,r^-_0(s,z))r^-_1(s,z)\right. \\ \quad&\qquad \qquad \qquad \qquad \left. +\frac{1}{2}\frac{\partial ^2 F^-_0}{\partial r^2}(s,x_0(s,z))(r^-_1(s,z))^2\right) ds\\&= \displaystyle \int \limits _0^{\pi }\left( \frac{F^+_2(s,r^+_0(s,z))}{\frac{\partial r_0^+(s,z)}{\partial z}}+\frac{\partial F^+_1}{\partial r}(s,r^+_0(s,z))v^1_1(\theta ,z)\right. \\ \quad&\qquad \qquad \qquad \left. +\frac{1}{2}\frac{\partial ^2 F^+_0}{\partial r^2}(s,x_0(s,z))(v^1_1(\theta ,z))^2\frac{\partial r_0^+(s,z)}{\partial z}\right) ds\\&\qquad \displaystyle \int \limits _\pi ^{2\pi }\left( \frac{F^-_2(s,r^-_0(s,z))}{\frac{\partial r_0^-(s,z)}{\partial z}}+\frac{\partial F^-_1}{\partial r}(s,r^-_0(s,z))v^1_2(\theta ,z)\right. \\ \quad&\qquad \qquad \qquad \left. +\frac{1}{2}\frac{\partial ^2 F^-_0}{\partial r^2}(s,x_0(s,z))(v^2_1(\theta ,z))^2\frac{\partial r_0^-(s,z)}{\partial z}\right) ds \end{aligned} \end{aligned}$$
(56)
Substituting (29) and (54)–(55) into (56), we can obtain the second order averaging function \(H_2(z)\). As we have seen, the concrete expression of \(H_2(z)\) will be much involved, and it is omitted.
