# American Institute of Mathematical Sciences

April  2019, 39(4): 1957-1974. doi: 10.3934/dcds.2019082

## Binary differential equations with symmetries

 1 Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil 2 Departamento de Matemática e Estatística, Universidade Federal S. J. del Rei, P. Frei Orlando, 170, Centro, S. J. del Rei, MG, 36307-352, Brazil

Received  February 2018 Revised  September 2018 Published  January 2019

Fund Project: This study was financed in part by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. Grant number 8474758/D for P.T. and CAPES/FCT grant 88887.125430/2016-00 for M.M.

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, $a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0,$ for $a, b, c$ smooth real functions defined on an open set of $\mathbb{R}^2$. Generically, solutions of a BDE are given as leaves of a pair of foliations, and the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts methods from invariant theory of compact Lie groups to obtain an algorithm to compute general expressions of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group ${{\bf{O}}(2)}$.

Citation: Miriam Manoel, Patrícia Tempesta. Binary differential equations with symmetries. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1957-1974. doi: 10.3934/dcds.2019082
##### References:

show all references

##### References:
Configurations of symmetric BDEs. In (a) and (b) the symmetry group is ${\bf Z}_2\times {\bf Z}_2$ and in (c) the symmetry group is ${\bf{D}}_6$
Configurations with symmetry (a) ${\bf{SO}}(2)$, (b) ${\bf{O}}(2)$ and (c) ${\bf{O}}(2)[{\bf{SO}}(2)]$
Configurations with symmetry group given by (a) ${\bf{Z}}_5$ and (b) ${\bf{Z}}_4[{\bf{Z}}_2]$
Configurations with symmetry groups ${\bf{D}}_5$, ${\bf{D}}_6[{\bf{Z}}_6]$ and ${\bf{D}}_6[{\bf{D}}_3(\kappa_x)]$
Configurations with symmetry groups ${\bf{Z}}_2$ and ${\bf{Z}}_2[{\bf 1}]$
Configurations with symmetry groups ${\bf{Z}}_2 \times {\bf{Z}}_2, {\bf{Z}}_2 \times {\bf{Z}}_2[{\bf{Z}}_2(-I)]$ and ${\bf{Z}}_2 \times {\bf{Z}}_2 [{\bf{Z}}_2(\kappa_x)]$
General forms of equivariant quadratic differential forms on the plane under closed subgroups of O$(2)$
 $\Gamma[\ker \eta]$ $\ker \lambda$ General form ${\bf{SO}}(2)$ ${\bf{SO}}(2)$ $a = p_1 + (y^2-x^2)p_2 + 2xyp_3;$ $b = 2xyp_2 + (x^2-y^2)p_3;$ $c = p_1 + (x^2-y^2)p_2 - 2xyp_3,$ $p_i \in \mathcal{P}({\bf{SO}}(2)), i = 1,2,3.$ ${\bf{O}}(2)$ ${\bf{SO}}(2)$ $a = p_1 + (y^2-x^2)p_2; \ b = 2xyp_2;$ $c = p_1 + (x^2-y^2)p_2, \ p_i \in \mathcal{P}({\bf{O}}(2)), i = 1,2.$ ${\bf{O}}(2)[{\bf{SO}}(2)]$ ${\bf{O}}(2)$ $a = 2xyp;$ $b = (x^2-y^2)p; $c = -2xyp, p \in \mathcal{P}({\bf{O}}(2)). {\bf Z}_n, \\ n \geq 3 {\bf Z}_n a = p_1 + (y^2-x^2)p_2 + 2xyp_3 -A_1p_4 -A_2p_5; b = 2xyp_2 + (x^2-y^2)p_3 + A_1p_5 -A_2p_4; c = p_1 + (x^2-y^2)p_2 -2xyp_3 +A_1p_4 + A_2p_5, p_i \in \mathcal{P}({\bf Z}_n ), i = 1,..., 5. {\bf Z}_n[{\bf Z}_{n/2}], n \geq 4 even {\bf Z}_{n/2} a = -A_3p_1-A_4p_2+A_5p_3-A_7p_4+A_8p_5+A_6p_6; b = -A_4p_1 + A_3p_2 + A_8p_4 + A_7p_5;$ c = A_3p_1 + A_4p_2 + A_5p_3 +A_7p_4 - A_8p_5 + A_6p_6, $p_i \in \mathcal{P}({\bf Z}_n), i = 1, ..., 6. {\bf{D}}_n,$ n \geq 3$ ${\bf Z}_n$ $a = p_1 + (y^2-x^2)p_2 - A_1p_3;$ $b = 2xyp_2 - A_2p_3; $c = p_1 + (x^2-y^2)p_2 + A_1p_3,$ p_i \in \mathcal{P}({\bf{D}}_n), i = 1,2,3.$ ${\bf{D}}_n[{\bf Z}_n],$ $n \geq 3$ ${\bf{D}}_n$ $a = 2xyp_1 -A_2p_2 +A_9p_3;$ $b = (x^2-y^2)p_1 + A_1p_2; $c = -2xyp_1 + A_2p_2 +A_9p_3,$ \ p_i \in \mathcal{P}({\bf{D}}_n), i = 1, 2, 3.$ ${\bf{D}}_n[{\bf{D}}_{n/2}(\kappa_x)], $n \geq 4 even {\bf{D}}_{n/2}(\kappa_y)] a = -A_3p_1 +A_5p_2 -A_7p_3; b = -A_4p_1 +A_8p_3; c = A_3p_1 +A_5p_2 + A_7p_3,$ p_i \in \mathcal{P}({\bf{D}}_n), i = 1, 2, 3.$ ${\bf Z}_2$ ${\bf 1}$ $a = p_1; \ b = yp_2; \ c = p_3, $p_i \in \mathcal{P}({\bf Z}_2), i = 1, 2,3. {\bf Z}_2[{\bf 1}] {\bf Z}_2 a = y p_1; \ b = p_2; \ c = yp_3,$ p_i \in \mathcal{P}({\bf Z}_2), i = 1, 2, 3.$ ${\bf Z}_2 \times {\bf Z}_2$ ${\bf Z}_2(-I)$ $a = p_1; \ b = xyp_2; \ c = p_3, $p_i \in \mathcal{P}({\bf Z}_2 \times {\bf Z}_2), i = 1, 2, 3. {\bf Z}_2 \times {\bf Z}_2[{\bf Z}_2(-I)] {\bf Z}_2 \times {\bf Z}_2 a = xyp_1; \ b = p_2; \ c = xyp_3,$ p_i \in \mathcal{P}({\bf Z}_2 \times {\bf Z}_2), i = 1, 2, 3.$ ${\bf Z}_2 \times {\bf Z}_2[{\bf Z}_2(\kappa_x)]$ ${\bf Z}_2(\kappa_y)$ $a = xp_1; \ b = yp_2; \ c = xp_3, $p_i \in \mathcal{P}({\bf Z}_2 \times {\bf Z}_2), i = 1, 2, 3. A_1 = \hbox{Re}(z^{n-2}), \ A_2 = \hbox{Im}(z^{n-2}), \ A_3 = \hbox{Re}(z^{n/2-2}), \ A_4 = \hbox{Im}(z^{n/2-2}), \ A_5 = \hbox{Re}(z^{n/2}), \ A_6 = \hbox{Im}(z^{n/2}), \ A_7 = \hbox{Re}(z^{n/2+2}), \ A_8 = \hbox{Im}(z^{n/2+2}), \ A_9 = \hbox{Im}(z^n).  \Gamma[\ker \eta] \ker \lambda General form {\bf{SO}}(2) {\bf{SO}}(2) a = p_1 + (y^2-x^2)p_2 + 2xyp_3; b = 2xyp_2 + (x^2-y^2)p_3; c = p_1 + (x^2-y^2)p_2 - 2xyp_3, p_i \in \mathcal{P}({\bf{SO}}(2)), i = 1,2,3. {\bf{O}}(2) {\bf{SO}}(2) a = p_1 + (y^2-x^2)p_2; \ b = 2xyp_2; c = p_1 + (x^2-y^2)p_2, \ p_i \in \mathcal{P}({\bf{O}}(2)), i = 1,2. {\bf{O}}(2)[{\bf{SO}}(2)] {\bf{O}}(2) a = 2xyp; b = (x^2-y^2)p;$ c = -2xyp, p \in \mathcal{P}({\bf{O}}(2)).$ ${\bf Z}_n, \\ n \geq 3$ ${\bf Z}_n$ $a = p_1 + (y^2-x^2)p_2 + 2xyp_3 -A_1p_4 -A_2p_5;$ $b = 2xyp_2 + (x^2-y^2)p_3 + A_1p_5 -A_2p_4;$ $c = p_1 + (x^2-y^2)p_2 -2xyp_3 +A_1p_4 + A_2p_5,$ $p_i \in \mathcal{P}({\bf Z}_n ), i = 1,..., 5.$ ${\bf Z}_n[{\bf Z}_{n/2}],$ $n \geq 4$ even ${\bf Z}_{n/2}$ $a = -A_3p_1-A_4p_2+A_5p_3-A_7p_4+A_8p_5+A_6p_6;$ $b = -A_4p_1 + A_3p_2 + A_8p_4 + A_7p_5; $c = A_3p_1 + A_4p_2 + A_5p_3 +A_7p_4 - A_8p_5 + A_6p_6,$ p_i \in \mathcal{P}({\bf Z}_n), i = 1, ..., 6.$ ${\bf{D}}_n, $n \geq 3 {\bf Z}_n a = p_1 + (y^2-x^2)p_2 - A_1p_3; b = 2xyp_2 - A_2p_3;$ c = p_1 + (x^2-y^2)p_2 + A_1p_3, $p_i \in \mathcal{P}({\bf{D}}_n), i = 1,2,3. {\bf{D}}_n[{\bf Z}_n], n \geq 3 {\bf{D}}_n a = 2xyp_1 -A_2p_2 +A_9p_3; b = (x^2-y^2)p_1 + A_1p_2;$ c = -2xyp_1 + A_2p_2 +A_9p_3, $\ p_i \in \mathcal{P}({\bf{D}}_n), i = 1, 2, 3. {\bf{D}}_n[{\bf{D}}_{n/2}(\kappa_x)],$ n \geq 4$ even ${\bf{D}}_{n/2}(\kappa_y)]$ $a = -A_3p_1 +A_5p_2 -A_7p_3;$ $b = -A_4p_1 +A_8p_3;$ $c = A_3p_1 +A_5p_2 + A_7p_3, $p_i \in \mathcal{P}({\bf{D}}_n), i = 1, 2, 3. {\bf Z}_2 {\bf 1} a = p_1; \ b = yp_2; \ c = p_3,$ p_i \in \mathcal{P}({\bf Z}_2), i = 1, 2,3.$ ${\bf Z}_2[{\bf 1}]$ ${\bf Z}_2$ $a = y p_1; \ b = p_2; \ c = yp_3, $p_i \in \mathcal{P}({\bf Z}_2), i = 1, 2, 3. {\bf Z}_2 \times {\bf Z}_2 {\bf Z}_2(-I) a = p_1; \ b = xyp_2; \ c = p_3,$ p_i \in \mathcal{P}({\bf Z}_2 \times {\bf Z}_2), i = 1, 2, 3.$ ${\bf Z}_2 \times {\bf Z}_2[{\bf Z}_2(-I)]$ ${\bf Z}_2 \times {\bf Z}_2$ $a = xyp_1; \ b = p_2; \ c = xyp_3, $p_i \in \mathcal{P}({\bf Z}_2 \times {\bf Z}_2), i = 1, 2, 3. {\bf Z}_2 \times {\bf Z}_2[{\bf Z}_2(\kappa_x)] {\bf Z}_2(\kappa_y) a = xp_1; \ b = yp_2; \ c = xp_3,$ p_i \in \mathcal{P}({\bf Z}_2 \times {\bf Z}_2), i = 1, 2, 3.$ $A_1 = \hbox{Re}(z^{n-2}), \ A_2 = \hbox{Im}(z^{n-2}), \ A_3 = \hbox{Re}(z^{n/2-2}), \ A_4 = \hbox{Im}(z^{n/2-2}), \ A_5 =$ $\hbox{Re}(z^{n/2}), \ A_6 = \hbox{Im}(z^{n/2}), \ A_7 = \hbox{Re}(z^{n/2+2}), \ A_8 = \hbox{Im}(z^{n/2+2}), \ A_9 = \hbox{Im}(z^n).$
 [1] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [2] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [3] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [4] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [5] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [6] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [7] Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374 [8] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [9] Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $\mathcal{W}(a, b, r)$. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123 [10] Lei Liu, Li Wu. Multiplicity of closed characteristics on $P$-symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378 [11] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [12] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [13] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [14] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [15] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [16] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [17] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [18] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [19] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [20] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

2019 Impact Factor: 1.338