# 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 and Continuous Dynamical Systems, 2019, 39 (4) : 1957-1974. doi: 10.3934/dcds.2019082
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##### References:
 [1] F. Antoneli, P. H. Baptistelli, A. P. S. Dias and M. Manoel, Invariant theory and reversible-equivariant vector fields, J. Pure Appl. Alg., 213 (2009), 649-663.  doi: 10.1016/j.jpaa.2008.08.002. [2] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer Verlag, New York, 1983. [3] J. W. Bruce and D. L. Fidal, On binary differential equations and umbilics, Proc. Roy. Soc. Edinburgh, Sect. A, 111 (1989), 147-168.  doi: 10.1017/S0308210500025087. [4] J. W. Bruce and F. Tari, On binary differential equations, Nonlinearity, 8 (1995), 255-271.  doi: 10.1088/0951-7715/8/2/008. [5] J. W. Bruce and F. Tari, Implicit differential equations from the singularity theory viewpoint, singularities and differential equations, Banach Center Publ., 33 (1996), 23-38. [6] J. W. Bruce and F. Tari, On the multiplicity of implicit differential equations, J. Diff. Eq., 148 (1998), 122-147.  doi: 10.1006/jdeq.1998.3454. [7] L. Dara, Singularités génériques des équations differentielles multiformes, Bull. Soc. Brazil Math., 6 (1975), 95-128.  doi: 10.1007/BF02584779. [8] G. Darboux, Sur la Forme Des Lignes de Courbure Dans la Voisignage D'un Ombilic, Lençons sur la Théorie des Surfaces, vol Ⅳ Note 7, Paris: Gautheir, Villares, (1896). [9] M. Field and M. Golubitsky, Symmetry in Chaos. A Search for Pattern in Mathematics, art, and Nature, 2nd edition, SIAM J. Appl. Math., Philadelphia, PA, 2009. [10] R. Garcia and J. Sotomayor, Differential Equations of Classical Geometry, a Qualitative Theory, 27a Colóquio Brasileiro de Matemática, Rio de Janeiro, 2009. [11] T. Golubitsky, I. Stewart and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol Ⅱ, Appl. Math. Sci. 69, Springer-Verlag, 1984. [12] V. Guíñez, Rank two codimention 1 singularities of positive quadratic differential equations, Nonlinearity, 10 (1997), 631-654.  doi: 10.1088/0951-7715/10/3/004. [13] V. Guíñez, Positive quadratic differential forms and foliations with singularities on surfaces, Trans. A.M.S., 309 (1998), 477-502.  doi: 10.1090/S0002-9947-1988-0961601-4. [14] Gutierrez,   and Guíñnez, Positive quadratic differential forms: Linearization, finite determinacy and versal unfolding, Ann. Fac. Sci. Toulouse Math, 5 (1996), 661-690.  doi: 10.5802/afst.844. [15] C. Gutierrez ans J. Sotomayor, Lines of curvature, umbilic points and Carathéodory conjecture, Resenhas IME-USP, 3 (1998), 291-322. [16] E. Hopf, Differential Geometry in the Large, L. N. M., 1000, Springer-Verlag, 1979. [17] M. Manoel and P. Tempesta, Homogeneus binary differential equations with symmetries, Preprint, 2018. [18] P. J. Olver, Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge, 1995. [19] J. Sotomayor and C. Gutierrez, Structurally stable configurations of lines of principal curvature, Bif. Erg. Th. Appl., Dijon, (1981), 195–215, Astérisque, (1982), 98–99. [20] F. Tari, Pairs of foliations on surfaces, in Proc. Real and Complex Singularities, LMS Lecture Notes Series, 380 (2010), 305–337. [21] F. Tari, Geometric properties of the solutions of implicit differential equations, Discrete Contin. Dyn. Syst., 17 (2007), 349-364.  doi: 10.3934/dcds.2007.17.349. [22] F. Tari, Two-parameter families of implicit differential equations, Discrete Contin. Dyn. Syst., 13 (2005), 139-162.  doi: 10.3934/dcds.2005.13.139.
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.$ $A_1 = \hbox{Re}(z^{n-2}), \ A_2 = \hbox{Im}(z^{n-2}), \ A_3 = \hbox{Re}(z^{n/2-2}), \ A_4 = \hbox{Im}(z^{n/2-2}), \ A_5 =$ $\hbox{Re}(z^{n/2}), \ A_6 = \hbox{Im}(z^{n/2}), \ A_7 = \hbox{Re}(z^{n/2+2}), \ A_8 = \hbox{Im}(z^{n/2+2}), \ A_9 = \hbox{Im}(z^n).$
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