Global Nilpotent Reversible Centers with Cubic Nonlinearities Symmetric with Respect to the x-Axis

Abstract

Let $P_3(x,y)$ and $Q_3(x,y)$ be polynomials of degree three without constant or linear terms. We characterize the global centers of all polynomial differential systems of the form $\dot{x} = y+ P_3(x,y)$, $\dot{y} =Q_3(x,y)$ that are reversible and invariant with respect to the x-axis.

Appendices

Appendix A Conditions in Theorem 1

Let

$$\begin{aligned} K^\pm {=}\frac{1}{27} \left( -2 a^3{+}3 a^2 d{+}9 a b{+}3 a d^2{-}18 b d{-}2 d^3\pm 2 \sqrt{\left( a^2-a d-3 b+d^2\right) ^3}\right) . \end{aligned}$$

The conditions in order that the origin is the unique finite singular point are the following ones:

$$\begin{aligned} c_1= & {} \{c=0,b=0,a=0\},\\ c_2= & {} \{c=0,b=0,a\ne 0,a\ge d\},\\ c_3= & {} \{c=0,b\ne 0,a^2-4b<0\},\\ c_4= & {} \{c=0,0<b\le a^2/4,d\le D^-\},\\ c_5= & {} \{c=0,b<0,d\le D^-\},\\ c_6= & {} \{c\ne 0,a=-d,b=d^2,c=-d^3\},\\ c_7= & {} \{c>0,a=c/d^2,b=-c/d,c\ne -d^3,d\ne 0\},\\ c_8= & {} \{c\ne 0, c=-bd,b\ne -ad, b>(a-d)^2/4\},\\ c_{9}= & {} \{c\ne 0,c=-bd, a> d,0< b \le (a-d)^2/4,d> 0 \},\\ c_{10}= & {} \{c\ne 0,c=-bd, a> -d, -a d<b \le (a-d)^2/4,d< 0 \},\\ c_{11}= & {} \{c\ne 0,c=-bd, 0< a< -d, -a d< b \le (a-d)^2/4,d< 0\}, \\ c_{12}= & {} \{c\ne 0,c=-bd, d< a \le 0,0< b \le (a-d)^2/4,d< 0 \},\\ c_{13}= & {} \{c\ne 0,c=-bd, a> 0, 0< b< -a d, d< 0 \}\\ c_{14}= & {} \{ c> 0,c + bd> 0, \Delta< 0\},\\ c_{15}= & {} \{ c> 0, c + bd< 0, \Delta< 0\},\\ c_{16}= & {} \{ a> 0, 0< b \le a^2/4,0< c< K^+|_{d=0},d = 0\},\\ c_{17}= & {} \{ a> 0, a^2/4< b< a^2/3, K^-|_{d=0}< c<K^+|_{d=0},d = 0\},\\ c_{18}= & {} \{a>-d, 0< b< -a d,-b d< c< K^+,d<0 \},\\ c_{19}= & {} \{a>-d,d(a-d)< b \le 0, 0< c< K^+,d<0\},\\ c_{20}= & {} \{a>-d,(a-d)^2/4< b< (a^2 - a d + d^2)/3, K^-< c< K^+,d<0 \},\\ c_{21}= & {} \{a>-d, -a d< b \le (a-d)^2/4, -b d< c< K^+, d<0 \},\\ c_{22}= & {} \{ a>2d, d(a - d)< b \le a^2/4,0< c< K^+, d>0\},\\ c_{23}= & {} \{a>2d, a^2/4< b< (a^2 - a d + d^2)/3, K^-< c< K^+,d>0\},\\ c_{24}= & {} \{d< a \le -d,0< b< (a-d)^2/4, -b d< c< K^+, d< 0\},\\ c_{25}= & {} \{ d< a \le -d, d(a-d)< b \le 0,0< c< K^+, d< 0 \},\\ c_{26}= & {} \{ a= -4 d, 0< b \le 4 d^2, 0< c< -b d, d< 0 \},\\ c_{27}= & {} \{ a= -4 d, 4d^2< b< 25d^2/4,K^-|_{a=-4d}< c< -b d, d< 0 \},\\ c_{28}= & {} \{ a=-d, 0< b \le d^2/4, 0< c< -b d, d< 0 \},\\ c_{29}= & {} \{ a=-d, d^2/4< b< d^2, K^-|_{a=-d}< c< -b d, d< 0 \},\\ c_{30}= & {} \{ a= d/2,0< b \le d^2/16, 0< c< -b d, d< 0 \},\\ c_{31}= & {} \{ a= d/2, d^2/16< b<d^2/4, K^-|_{a=d/2}< c< K^+|_{a=d/2}, d< 0 \},\\ c_{32}= & {} \{ a=d, 0< b \le d^2/4, 0< c< K^+|_{a=d}, d< 0 \},\\ c_{33}= & {} \{ a=d, d^2/4< b< d^2/3, K^-|_{a=d}< c< K^+|_{a=d}, d< 0 \},\\ c_{34}= & {} \{ a> -4 d, 0< b \le a^2/4, 0< c< -b d, d< 0 \},\\ c_{35}= & {} \{ a> -4 d, a^2/4< b< (a-d)^2/4, K^-< c< -b d, d< 0 \},\\ c_{36}= & {} \{ 0< a< -d, 0< b \le a^2/4, 0< c< -b d, d< 0 \},\\ c_{37}= & {} \{ 0< a< -d, a^2/4< b< -a d,K^-< c< -b d, d< 0 \},\\ c_{38}= & {} \{ 0< a< -d, (a-d)^2/4< b< (a^2 - a d + d^2)/3, K^-< c< K^+, d< 0 \},\\ c_{39}= & {} \{ 0< a< -d, -a d< b \le (a-d)^2/4, K^-< c< -b d, d< 0 \},\\ c_{40}= & {} \{-d< a< -4 d, 0< b \le a^2/4, 0< c< -b d, d< 0 \},\\ c_{41}= & {} \{ d< a< d/2, 0< b \le (a-d)^2/4, 0< c< -b d, d< 0 \},\\ c_{42}= & {} \{ d/2< a \le 0, 0< b \le a^2/4, 0< c< -b d, d< 0 \},\\ c_{43}= & {} \{ d< a< d/2, (a-d)^2/4< b \le a^2/4, 0< c< K^+, d< 0 \},\\ c_{44}= & {} \{ 2 d< a< d, d(a-d)< b \le a^2/4,0< c< K^+, d< 0 \},\\ c_{45}= & {} \{ -d< a< -4 d, a^2/4< b< (a-d)^2/4, K^-< c< -b d, d< 0 \},\\ c_{46}= & {} \{ d< a< d/2, a^2/4< b< (a^2 - a d + d^2)/3, K^-< c< K^+, d< 0 \},\\ c_{47}= & {} \{ 2 d< a< d, a^2/4< b< (a^2 - a d + d^2)/3, K^-< c< K^+, d< 0 \},\\ c_{48}= & {} \{ d/2< a \le 0, a^2/4< b \le (a-d)^2/4, K^-< c< -b d, d< 0 \},\\ c_{49}= & {} \{d/2< a \le 0, (a-d)^2/4< b< (a^2 - a d + d^2)/3, K^-< c< K^+, d< 0 \},\\ c_{50}= & {} \{ a>-d,b=(a^2-ad+d^2)/3,c=(a-2d)^3/27,d\le 0\},\\ c_{51}= & {} \{a>2d,b=(a^2-ad+d^2)/3,c=(a-2d)^3/27,d> 0 \},\\ c_{52}= & {} \{2d<a<-d,b=(a^2-ad+d^2)/3,c=(a-2d)^3/27,d< 0\},\\ c_{53}= & {} \{a> 0, a^2/4< b< a^2/3,c = K^-|_{d=0}, d = 0 \},\\ c_{54}= & {} \{a> 0,0< b \le a^2/4, c = K^+|_{d=0}, d = 0 \},\\ c_{55}= & {} \{a> 0, a^2/4< b< a^2/3, c = K^+|_{d=0}, d = 0\},\\ c_{56}= & {} \{ a> -d, (a-d)^2/4< b< (a^2 - a d + d^2)/3, c= K^-,d< 0 \},\\ c_{57}= & {} \{ a> 2 d,a^2/4< b< (a^2 - a d + d^2)/3, c=K^-, d> 0\},\\ c_{58}= & {} \{a> -d, d(a-d)< b< -a d, c= K^+, d< 0 \},\\ c_{59}= & {} \{ a> -d, (a-d)^2/4< b< (a^2 - a d + d^2)/3, c=K^+, d< 0 \},\\ c_{60}= & {} \{ a> -d, -a d< b \le (a-d)^2/4, c=K^+, d< 0 \},\\ c_{61}= & {} \{ a> 2 d, a^2/4< b< (a^2 - a d + d^2)/3,c=K^+,d> 0 \},\\ c_{62}= & {} \{ a> 2 d, d(a-d)< b \le a^2/4, c=K^+, d> 0\},\\ c_{63}= & {} \{ d< a \le -d, d(a-d)< b< (a-d)^2/4, c=K^+, d< 0 \},\\ c_{64}= & {} \{a=-4 d, 4d^2< b< 25d^2/4,c=K^-|_{a=-4d}, d< 0 \},\\ c_{65}= & {} \{ a=-d, d^2/4< b< d^2, c=K^-|_{a=-d}, d< 0\},\\ c_{66}= & {} \{ a=d/2, d^2/16< b< d^2/4, c=K^-|_{a=d/2}, d< 0 \},\\ c_{67}= & {} \{ a=d/2, d^2/16< b< d^2/4, c=K^+|_{a=d/2}, d< 0 \},\\ c_{68}= & {} \{ a = d, d^2/4< b< d^2/3, c=K^-|_{a=d}, d< 0\},\\ c_{69}= & {} \{ a = d, d^2/4< b< d^2/3,c=K^+|_{a=d}, d< 0\},\\ c_{70}= & {} \{a = d, 0< b \le d^2/4, c=K^+|_{a=d}, d< 0\},\\ c_{71}= & {} \{ a > -4 d, a^2/4< b< (a-d)^2/4, c=K^-, d< 0\},\\ c_{72}= & {} \{ 0< a< -d, a^2/4< b< -a d, c=K^-, d< 0\},\\ c_{73}= & {} \{ 0< a< -d, (a-d)^2/4< b< (a^2 - a d + d^2)/3,c=K^-,d< 0 \},\\ c_{74}= & {} \{ 0< a< -d, -a d< b \le (a-d)^2/4, c=K^-,d< 0 \},\\ c_{75}= & {} \{ -d< a< -4 d, a^2/4< b< (a-d)^2/4, c=K^-, d< 0\},\\ c_{76}= & {} \{ d< a< d/2, a^2/4< b< (a^2 - a d + d^2)/3, c=K^-,d< 0 \},\\ c_{77}= & {} \{ 2 d< a< d, a^2/4< b< (a^2 - a d + d^2)/3,c=K^-, d< 0 \},\\ c_{78}= & {} \{ d/2< a \le 0, a^2/4< b \le (a-d)^2/4, c=K^-, d< 0 \},\\ c_{79}= & {} \{ d/2< a \le 0, (a-d)^2/4< b< (a^2 - a d + d^2)/3, c=K^-, d< 0 \},\\ c_{80}= & {} \{ 0< a< -d, (a-d)^2/4< b< (a^2 - a d + d^2)/3,c=K^+,d< 0 \},\\ c_{81}= & {} \{ d< a< d/2, a^2/4< b< (a^2 - a d + d^2)/3, c=K^+,d< 0 \},\\ c_{82}= & {} \{ d< a< d/2, (a-d)^2/4< b \le a^2/4, c=K^+, d< 0 \},\\ c_{83}= & {} \{ 2 d< a< d, a^2/4< b< (a^2 - a d + d^2)/3,c=K^+, d< 0 \},\\ c_{84}= & {} \{ 2 d< a< d, d(a-d)< b \le a^2/4,c=K^+,d< 0 \},\\ c_{85}= & {} \{ d/2< a \le 0, (a-d)^2/4< b< (a^2 - a d + d^2)/3, c=K^+, d < 0 \}. \end{aligned}$$

The conditions in order that the local chart $U_1$ has either no infinite singular points or all the infinite singular points in the local chart $U_1$ are formed by two degenerated hyperbolic sectors are the following:

$$\begin{aligned} i_1= & {} \{c=0, b=d\},\\ i_2= & {} \{c=0,b>d\},\\ i_3= & {} \{c\ne 0,b^2-4 c-2 b d+d^2<0\},\\ i_4= & {} \{c\ne 0, b> d,0 < c \le (b-d)^2/4\},\\ i_5= & {} \{c\ne 0,a=1,b=-d,c=d^2,d>1/8\}. \end{aligned}$$

The conditions in order that the origin of the local chart $U_2$ is either not a singular point or it is formed by two degenerated hyperbolic sectors are:

$$\begin{aligned} j_1= & {} \{c\ne 0\},\\ j_2= & {} \{c=0, a^2-4b<0, b>d\},\\ j_3= & {} \{c=0,a^2-4b<0,b=d,d>0\},\\ j_4= & {} \{c=0,a=b=0,d<0\}. \end{aligned}$$

The sets $ij_i$ for $i=1,\dots ,6$ are:

$$\begin{aligned} ij_1= & {} \{b>a^2/4,b=d,c=0\},\\ ij_2= & {} \{a=0,b=0,c=0,d<0\},\\ ij_3= & {} \{b>a^2/4,c=0,d<b\},\\ ij_4= & {} \{a=1,b=-d,c=d^2,d>1/8\},\\ ij_5= & {} \{b>d,0<c\le (b-d)^2/4\},\\ ij_6= & {} \{c> (b-d)^2/4\}. \end{aligned}$$

The sets $d_i$ in Theorem 1 are:

$$\begin{aligned} d_{1}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{3}}\cap {ij_{1}}\}\ni \left( 0,1,0,1\right) ,\\ d_{2}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{1}}\cap {ij_{2}}\}\ni \left( 0,0,0,-1\right) ,\\ d_{3}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{3}}\cap {ij_{3}}\}\ni \left( 0, 1, 0, 1/2\right) ,\\ d_{4}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{7}}\cap {ij_{4}}\}\ni \left( 1, -1, 1, 1\right) ,\\ d_{5}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{6}}\cap {ij_{5}}\}\ni \left( 2,4,8,-2\right) ,\\ d_{6}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{7}}\cap {ij_{5}}\}\ni \left( 1/2,1/8,1/32,-1/4\right) ,\\ d_{7}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{8}}\cap {ij_{5}}\}\ni \left( 0,3/2,3,-2\right) ,\\ d_{8}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{10}}\cap {ij_{5}}\}\ni \left( 2,17/8,17/8,-1\right) ,\\ d_{9}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{11}}\cap {ij_{5}}\}\ni \left( 1/2,3,15,-5\right) ,\\ d_{10}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{12}}\cap {ij_{5}}\}\ni \left( -1,5/2,25/2,-5\right) ,\\ d_{11}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{13}}\cap {ij_{5}}\}\ni \left( 1,1/2,1/2,-1\right) ,\\ d_{12}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{14}}\cap {ij_{5}}\}\ni \left( -3,17/8,161/512,1\right) ,\\ d_{13}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{15}}\cap {ij_{5}}\}\ni \left( -2333/8,137,280725/8,-269\right) ,\\ d_{14}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{16}}\cap {ij_{5}}\}\ni \left( 1/2,1/32,1/8192,0\right) ,\\ d_{15}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{17}}\cap {ij_{5}}\}\ni \left( 1/2,133/2048,27/32768,0\right) ,\\ d_{16}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{18}}\cap {ij_{5}}\}\ni \left( 3/2,1/4,5/16,-1\right) ,\\ d_{17}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{19}}\cap {ij_{5}}\}\ni \left( 7/4,-1/2,1/32,-1\right) ,\\ d_{18}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{20}}\cap {ij_{5}}\}\ni \left( 7/4,245/128,985/512,-1\right) ,\\ d_{19}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{21}}\cap {ij_{5}}\}\ni \left( 3/2,49/32,6273/4096,-1\right) ,\\ d_{20}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{22}}\cap {ij_{5}}\}\ni \left( 13/4,39/16,1/256,1\right) ,\\ d_{21}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{23}}\cap {ij_{5}}\}\ni \left( 13/4,173/64,45/1024,1\right) ,\\ d_{22}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{24}}\cap {ij_{5}}\}\ni \left( -27/32,3/1024,9/2048,-1\right) ,\\ d_{23}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{25}}\cap {ij_{5}}\}\ni \left( -1/2,-1/4,3/512,-1\right) ,\\ d_{24}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{26}}\cap {ij_{5}}\}\ni \left( 4,1/2,1/4,-1\right) ,\\ d_{25}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{27}}\cap {ij_{5}}\}\ni \left( 4,37/8,3,-1\right) ,\\ d_{26}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{28}}\cap {ij_{5}}\}\ni \left( 5,3,29/2,-5\right) ,\\ d_{27}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{29}}\cap {ij_{5}}\}\ni \left( 1,11/32,11/64,-1\right) ,\\ d_{28}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{30}}\cap {ij_{5}}\}\ni \left( -1/2,1/32,1/64,-1\right) ,\\ d_{29}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{31}}\cap {ij_{5}}\}\ni \left( -1/2,13/128,5/128,-1\right) ,\\ \end{aligned}$$

$$\begin{aligned} d_{30}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{32}}\cap {ij_{5}}\}\ni \left( -4,2,1/8,-4\right) ,\\ d_{31}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{33}}\cap {ij_{5}}\}\ni \left( -4,37/8,23/16,-4\right) ,\\ d_{32}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{34}}\cap {ij_{5}}\}\ni \left( 5,1/2,1/4,-1\right) ,\\ d_{33}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{35}}\cap {ij_{5}}\}\ni \left( 8,18,14,-1\right) ,\\ d_{34}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{36}}\cap {ij_{5}}\}\ni \left( 5/2,1,5/2,-3\right) ,\\ d_{35}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{37}}\cap {ij_{5}}\}\ni \left( 1/8,1/4,511/512,-4\right) ,\\ d_{36}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{38}}\cap {ij_{5}}\}\ni \left( 5/32,39/8,73/4,-4\right) ,\\ d_{37}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{39}}\cap {ij_{5}}\}\ni \left( 1/8,9/4,569/64,-4\right) ,\\ d_{38}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{40}}\cap {ij_{5}}\}\ni \left( 5/4,1/4,1/8,-1\right) ,\\ d_{39}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{41}}\cap {ij_{5}}\}\ni \left( -12,7/2,59,-17\right) ,\\ d_{40}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{42}}\cap {ij_{5}}\}\ni \left( -9,21/2,220,-21\right) ,\\ d_{41}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{43}}\cap {ij_{5}}\}\ni \left( -16,53,648,-29\right) ,\\ d_{42}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{44}}\cap {ij_{5}}\}\ni \left( -4,7/2,1/64,-3\right) ,\\ d_{43}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{45}}\cap {ij_{5}}\}\ni \left( 3/2,25/32,19/32,-1\right) ,\\ d_{44}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{46}}\cap {ij_{5}}\}\ni \left( -3/4,13/64,45/1024,-1\right) ,\\ d_{45}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{47}}\cap {ij_{5}}\}\ni \left( -4,17/4,45/128,-13/4\right) ,\\ d_{46}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{48}}\cap {ij_{5}}\}\ni \left( -1/4,5/64,17/256,-1\right) ,\\ d_{47}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{49}}\cap {ij_{5}}\}\ni \left( -1/4,13/64,163/1024,-1\right) ,\\ d_{48}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{50}}\cap {ij_{5}}\}\ni \left( 2,7/3,64/27,-1\right) ,\\ d_{49}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{51}}\cap {ij_{5}}\}\ni \left( 3,7/3,1/27,1\right) ,\\ d_{50}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{52}}\cap {ij_{5}}\}\ni \left( -2,4/3,8/27,-2\right) ,\\ d_{51}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{53}}\cap {ij_{5}}\}\\{} & {} \ni \left( 1/2,133/2048,\left( 2768-113 \sqrt{226}\right) /1769472,0\right) ,\\ d_{52}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{54}}\cap {ij_{5}}\}\\{} & {} \ni \left( 35/32,3/32,\left( -27755+937 \sqrt{937}\right) /442368,0\right) ,\\ d_{53}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{55}}\cap {ij_{5}}\}\ni \left( 81/64,57/128,49/1024,0\right) ,\\ d_{54}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{56}}\cap {ij_{5}}\}\ni \left( 7/4,245/128,\left( 1970-\sqrt{2}\right) /1024,-1\right) ,\\ d_{55}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{57}}\cap {ij_{5}}\}\ni \left( 4,133/32,\left( 616-17 \sqrt{34}\right) /3456,1\right) ,\\ d_{56}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{58}}\cap {ij_{5}}\}\ni \left( 7/4,25/16,\left( 95+4 \sqrt{2}\right) /64,-1\right) ,\\ d_{57}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{59}}\cap {ij_{5}}\}\ni \left( 7/4,245/128,\left( 1970+\sqrt{2}\right) /1024,-1\right) ,\\ d_{58}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{60}}\cap {ij_{5}}\}\ni \left( 7/4,29/16,\left( 1035+4 \sqrt{6}\right) /576,-1\right) ,\\ \end{aligned}$$

$$\begin{aligned}d_{59}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{61}}\cap {ij_{5}}\}\ni \left( 4,133/32,\left( 616+17 \sqrt{34}\right) /3456,1\right) ,\\ d_{60}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{62}}\cap {ij_{5}}\}\ni \left( 4,7/2,\left( -14+5 \sqrt{10}\right) /54,1\right) ,\\ d_{61}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{63}}\cap {ij_{5}}\}\ni \left( -3/4,-1/8,\left( -80+19 \sqrt{19}\right) /864,-1\right) ,\\ d_{62}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{64}}\cap {ij_{5}}\}\ni \left( 4,5,\left( 36-4 \sqrt{6}\right) /9,-1\right) ,\\ d_{63}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{65}}\cap {ij_{5}}\}\ni \left( 1/4,5/128,\left( 10-\sqrt{2}\right) /1024,-1/4\right) ,\\ d_{64}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{66}}\cap {ij_{5}}\}\ni \left( -16,160,2560-256 \sqrt{2},-32\right) ,\\ d_{65}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{67}}\cap {ij_{5}}\}\ni \left( -16,160,2560+256 \sqrt{2},-32\right) ,\\ d_{66}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{68}}\cap {ij_{5}}\}\ni \left( -1,37/128,\left( 616-17 \sqrt{34}\right) /27648,-1\right) ,\\ d_{67}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{69}}\cap {ij_{5}}\}\ni \left( -1,37/128,\left( 616+17 \sqrt{34}\right) /27648,-1\right) ,\\ d_{68}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{70}}\cap {ij_{5}}\}\ni \left( -1,1/8, \left( -14+5 \sqrt{10}\right) /432,-1\right) ,\\ d_{69}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{71}}\cap {ij_{5}}\}\ni \left( 8,18,\left( 430-38 \sqrt{19}\right) /27,-1\right) ,\\ d_{70}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{72}}\cap {ij_{5}}\}\ni \left( 1/8,1/8,\left( 1881-83 \sqrt{249}\right) /2304,-2\right) ,\\ d_{71}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{73}}\cap {ij_{5}}\}\\{} & {} \ni \left( 5/32,39/8,\left( 2692737-691 \sqrt{2073}\right) /147456,-4\right) ,\\ d_{72}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{74}}\cap {ij_{5}}\}\ni \left( 1/8,9/4,1125/128,-4\right) ,\\ d_{73}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{75}}\cap {ij_{5}}\}\ni \left( 7/4,17/16,\left( 495-28 \sqrt{42}\right) /576,-1\right) ,\\ d_{74}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{76}}\cap {ij_{5}}\}\ni \left( -16,118,1080-108 \sqrt{2},-26\right) ,\\ d_{75}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{77}}\cap {ij_{5}}\}\ni \left( -4,17/4,\left( 305-13 \sqrt{13}\right) /864,-13/4\right) ,\\ d_{76}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{78}}\cap {ij_{5}}\}\ni \left( -1/4,5/64,\left( 595-37 \sqrt{37}\right) /6912,-1\right) ,\\ d_{77}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{79}}\cap {ij_{5}}\}\ni \left( -1/4,13/64,\left( 1099-13 \sqrt{13}\right) /6912,-1\right) ,\\ d_{78}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{80}}\cap {ij_{5}}\}\\{} & {} \ni \left( 7/32,51/128,\left( 171181+73 \sqrt{73}\right) /442368,-1\right) ,\\ d_{79}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{81}}\cap {ij_{5}}\}\ni \left( -16,118,1080+108 \sqrt{2},-26\right) ,\\ d_{80}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{82}}\cap {ij_{5}}\}\ni \left( -3/4,5/64,\left( -55+37 \sqrt{37}\right) /6912,-1\right) ,\\ d_{81}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{83}}\cap {ij_{5}}\}\ni \left( -4,17/4, \left( 305+13 \sqrt{13}\right) /864,-13/4\right) ,\\ d_{82}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{84}}\cap {ij_{5}}\}\ni \left( -4,7/2, \left( -14+5 \sqrt{10}\right) /54,-3\right) ,\\ d_{83}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{85}}\cap {ij_{5}}\}\ni \left( -1/4,13/64,\left( 1099+13 \sqrt{13}\right) /6912,-1\right) ,\\ \end{aligned}$$

$$\begin{aligned} d_{84}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{14}}\cap {ij_{6}}\}\ni \left( -5,-70,1280,1\right) ,\\ d_{85}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{16}}\cap {ij_{6}}\}\ni \left( 1/2,1/32,13/32768,0\right) ,\\ d_{86}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{17}}\cap {ij_{6}}\}\ni \left( 1/2,133/2048,15/8192,0\right) ,\\ d_{87}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{18}}\cap {ij_{6}}\}\ni \left( 3/2,1/4,1/2,-1\right) ,\\ d_{88}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{19}}\cap {ij_{6}}\}\ni \left( 7/4,-21/16,11/128,-1\right) ,\\ d_{89}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{20}}\cap {ij_{6}}\}\ni \left( 1,53831/65536,87485/131072,-13/16\right) ,\\ d_{90}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{21}}\cap {ij_{6}}\}\ni \left( 1,209/256,347783/524288,-13/16\right) ,\\ d_{91}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{22}}\cap {ij_{6}}\}\ni \left( 5/16,19/2048,1/65536,3/256\right) ,\\ d_{92}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{23}}\cap {ij_{6}}\}\ni \left( 5/16,805/32768,1/4096,3/256\right) ,\\ d_{93}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{24}}\cap {ij_{6}}\}\ni \left( 29/64,7/512,1055/4096,-1\right) ,\\ d_{94}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{25}}\cap {ij_{6}}\}\ni \left( 7/32,-67/64,1/512,-1\right) ,\\ d_{95}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{50}}\cap {ij_{6}}\}\ni \left( 1/2,7/64,1/64,-1/8\right) ,\\ d_{96}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{51}}\cap {ij_{6}}\}\ni \left( 1/2,19/256,1/512,1/16\right) ,\\ d_{97}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{53}}\cap {ij_{6}}\}\ni \left( 1/2,39/512,(760-11 \sqrt{22})/221184,0\right) ,\\ d_{98}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{54}}\cap {ij_{6}}\}\ni \left( 1/2,1/32,(-14+5 \sqrt{10})/3456,0\right) ,\\ d_{99}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{55}}\cap {ij_{6}}\}\\{} & {} \ni \left( 1/2,133/2048,(2768+113 \sqrt{226})/1769472,0\right) ,\\ d_{100}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{56}}\cap {ij_{6}}\}\\{} & {} \ni \left( 1,1685/2048,(43820-\sqrt{2})/65536,-13/16\right) ,\\ d_{101}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{57}}\cap {ij_{6}}\}\ni \left( 1,37/128,121/8192,3/32\right) ,\\ d_{102}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{58}}\cap {ij_{6}}\}\ni \left( 8,0, (-1280+896 \sqrt{7})/27,-4\right) ,\\ d_{103}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{59}}\cap {ij_{6}}\}\\{} & {} \ni \left( 5/32,589/65536,(148715+217 \sqrt{217})/536870912,-25/1024\right) ,\\ d_{104}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{60}}\cap {ij_{6}}\}\\{} & {} \ni \left( 1/8,75/16384,(7652195+15577 \sqrt{15577})/115964116992,-27/2048\right) ,\\ d_{105}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{61}}\cap {ij_{6}}\}\\{} & {} \ni \left( 5/16,805/32768,(53983+1339 \sqrt{1339})/226492416,3/256\right) ,\\ d_{106}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{62}}\cap {ij_{6}}\}\\{} & {} \ni \left( 5/16,1/64,(-141155+3097 \sqrt{3097})/226492416,3/256\right) ,\\ d_{107}= & {} \{(a,b,c,d)\in \mathbb {R}^{4}\,:\,\mathrm {c_{63}}\cap {ij_{6}}\}\ni \left( -9/2,0,(-575+193 \sqrt{193})/108,-8\right) . \end{aligned}$$

Appendix B Conditions in Theorem 2

The conditions in order that the origin is the unique finite singular point are:

$$\begin{aligned} C_1= & {} \{b \ge 0, c \le 0\}, \\ C_2= & {} \left\{ b \ge 0, c>0,a>\frac{c-4 b}{4 c}\right\} . \end{aligned}$$

The conditions in order that the local chart $U_1$ has either no infinite singular points or all the infinite singular points in the local chart $U_1$ are formed by two degenerated hyperbolic sectors are the following:

$$\begin{aligned} I_1= & {} \{b=0,a=c\},\\ I_2= & {} \{b=0,a>c\},\\ I_3= & {} \{b\ne 0, b>(a-c)^2/4 \},\\ I_4= & {} \{b\ne 0, a>c, 0<b \le (a-c)^2/4 \}. \end{aligned}$$

The conditions in order that the origin of the local chart $U_2$ is either not a singular point or it is formed by two degenerated hyperbolic sectors are:

$$\begin{aligned} J_1= & {} \{b\ne 0\},\\ J_2= & {} \{b= 0,a>1/4,a>c\},\\ J_3= & {} \{b=0,a=c>1/4\}. \end{aligned}$$

The sets $IJ_i$ for $i=1,\dots 5$ are:

$$\begin{aligned} IJ_1= & {} \{ b> (a-c)^2/4\},\\ IJ_2= & {} \{b>0, a> c\},\\ IJ_3= & {} \{b = 0,a=c> 1/4\},\\ IJ_4= & {} \{b = 0, a> 1/4, c \le 1/4\},\\ IJ_5= & {} \{b = 0, a> c > 1/4\}. \end{aligned}$$

The conditions in Theorem 2 are:

$$\begin{aligned} e_{1}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_1}\cap IJ_1\}\ni \left( 0,2,-1\right) ,\\ e_{2}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_1}\cap IJ_2\}\ni \left( 0,1,-1\right) ,\\ e_{3}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_1}\cap IJ_4\}\ni \left( 5/4,0,0\right) ,\\ e_{4}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_2}\cap IJ_1\}\ni \left( 0,2,1\right) ,\\ e_{5}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_2}\cap IJ_2\}\ni \left( 2,1,1\right) ,\\ e_{6}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_2}\cap IJ_3\}\ni \left( 1,0,1\right) ,\\ e_{7}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_2}\cap IJ_4\}\ni \left( 1,0,1/8\right) ,\\ e_{8}= & {} \{(a,b,c)\in \mathbb {R}^3\,:\,\mathrm {C_2}\cap IJ_5\}\ni \left( 2,0,1\right) . \end{aligned}$$